P.L.Spheres, convex polytopes, and stress
Discrete Comput Geom
P.L.Spheres 0
Convex Polytopes 0
Stress 0
0 Department of Mathematics,University of Kentucky , Lexington, KY 40506 , USA
We describe here the notion of generalized stress on simplicial complexes, which serves several purposes: it establishes a link between two proofs oftbe Lower Bound Theorem for simplicial convex polytopes; elucidates some connections between the algebraic tools and the geometric properties of polytopes; leads to an associated natural generalization of infinitesimal motions; behaves well with respect to bistellar operations in the same way that the face ring of a simplicial complex coordinates well with shelling operations, giving rise to a new proof that p.1.spheres are CohenMacaulay; and is dual to the notion of McMullen's weights on simple polytopes which he used to give a simpler, more geometric proof of the gtheorem. * Supported in part by NSF Grants DMS8504050 and DMS8802933, by NSA Grant MDA90489H2038, by the MittagLefflerInstitute,by DIMACS(Center for DiscreteMathematicsandTheoretical Computer Science),a National Science Foundation Science and Technology Center, NSFSTC8809648,and by a grant from the EPSRC.

C. W. Lee
1. Introduction
A b o u t 25 years ago two important extremal p r o b l e m s for convex polytopes were solved
at almost the same time. M c M u l l e n [
12
] proved the U p p e r Bound Conjecture, which
predicts the m a x i m u m n u m b e r of faces of each dimension that a convex d  p o l y t o p e
(ddimensional polytope) with n vertices can have. Barnette [
1
], [
2
] settled the L o w e r
Bound Conjecture, which specifies the m i n i m u m number o f faces of each d i m e n s i o n
that a simplicial convex d  p o l y t o p e with n vertices can possess.
The first proofs o f these results were s o m e w h a t unrelated, but in the subsequent
decade Stanley developed a c o m m o n algebraic perspective for recasting and ultimately
reproving both of these results [
21
]. In fact, he established the complete characterization
of facevectors of simplicial (or dually, simple) polytopes (the g  t h e o r e m ) originally
conjectured by McMullen. This quickly led to the further development of very powerful
connections between the combinatorics of convex polytopes and the algebraic geometry
of associated toric varieties [
6
], [
16
]. This interplay has proved to be very fruitful and
far from exhausted, and many issues are as yet unresolved. Some important progress,
as well as questions, center around the extensions of the facecounting results to other
classes of objects such as nonsimplicial (or nonsimple) polytopes, unbounded polyhedra,
or simplicial spheres.
We describe here the notion of generalized stress, which serves several purposes: it
establishes a link between two proofs of the Lower Bound Theorem; elucidates some
connections between the algebraic tools and the geometric properties of polytopes; leads
to an associated natural generalization of infinitesimal motions; behaves well with
respect to bistellar operations in the same way that the face ring of a simplicial complex
coordinates well with shelling operations, giving rise to a new proof that p.1.spheres are
CohenMacaulay; and is dual to the notion of McMullen's weights on simple polytopes
which he used to give a simpler, more geometric proof of the gtheorem [
13
], [
14
].
Generalized stress was first introduced in [
10
], and a detailed overview was presented
in [
11
].
2.
The L o w e r B o u n d T h e o r e m
For a simplicial convex dpolytope P, let
g2 = f ,  d f o + ( d + l )
2
"
We begin by sketching two proofs of the Lower Bound Theorem, which states:
T h e o r e m 1 (Barnette). For all simplicial polytopes, g2 is nonnegative.
Here, fj denotes the number of jfaces (jdimensional faces) of P. The first proof is
due to Stanley, the second to Kalai.
2.1. Stanley's Proof
Stanley's [
21
] proof of this result is actually an easy corollary of his proof of the more
powerful gtheorem, and requires some preliminary definitions. Let A be a simplicial
(d  1)complex ((d  1)dimensional complex) on the vertex set {1. . . . . n }. The f vector
of A is the vector of nonnegative integers f = (f0 . . . . . fa1), where 3~ denotes the
number of faces (elements) of A of dimension j (cardinality j + 1). With the convention
that f1 = 1, the hvector of A is the vector of integers h = (h0 . . . . . ha) defined by
hk
~
(1) jk ( ; )J
 k
As is well known, the hvector encodes the same amount o f information as the fvector,
since
J3=
J + ~ ( d  k
d  j  1
) h k ,
j =   I . . . . . d   1 .
k=0
N o w define go = h0 = 1 and gk hk  hk1, k = 1. . . . . [ d / 2 J .
The face ring of A over R is A = R[Xl . . . . . x n ] / l a , where It, is the ideal generated
by all squarefree monomials xi~ . . . xi, such that {il . . . . . is} is not a m e m b e r o f A. We
grade A in a natural way by degree, A = A0 ~3 AI ~B A2 ~ 99. For 01 . . . . . Od~ AI,
define B = B0 ~ Bl fl) . . . . A/(01 . . . . . Od). Stanley [
19
], [
20
] proves:
Theorem 2 (Stanley). A is CohenMacaulay if and only if Or. . . . . Od exist such that
B = Bo ~ . . . ~ Bd and dim Bk = hk, k ~0 . . . . . d. In this case the Oj can be chosen
generically (i.e., with coefficients that are algebraically independent over R).
I f the ring A is CohenMacaulay, then A is called a CohenMacaulay complex.
Reisner [
18
] gives a homological characterization o f the class of C o h e n  M a c a u l a y
complexes, which includes shellable simplicial complexes, simplicial balls and spheres, and
boundary complexes o f simplicial polytopes. The hvectors o f C o h e n  M a c a u l a y
complexes are clearly nonnegative, but they must also satisfy certain nonlinear conditions.
For01 . . . . . Od ~ A1, B = A / (01 . . . . . Od), and oo ~ B1, define C = Co ~BC1 ~3 . . . .
B/(og). Stanley exploits a connection between the face ring o f a simplicial convex
polytope and the cohomology o f an associated toric variety, and invokes the Hard Lefschetz
Theorem for such varieties to prove:
T h e o r e m 3 (Stanley). Suppose that A is the face ring of the boundary complex A of
some simplicial convex dpolytope. Then O1. . . . . Od E A 1 and co ~ Bl exist such that:
1. B = B o ~ 3 .  . ~ Bd a n d d i m B k = hk, k = O . . . . . d.
2. Multiplication by coa2k is a bijection between Bk and Bak, k = 0 . . . . . Ld/2J.
In particular, multiplication by o9is an injection from Bk into Bk+t, k = 0 . . . . . Ld/2J  1.
As a consequence, C = Co fl) . . . ~ CLd/2j and gk = dim Ck, k = 0 . . . . . Ld/2J.
A n immediate corollary is that the numbers gk are nonnegative, k = 0 . . . . . [ d / 2 ] .
(This was first conjectured by McMullen and Walkup [
15
].) In particular, g2 > 0. We
also see that h i ~ hai, i = 0 . . . . . [ d / 2 J . These are the DehnSommerville relations,
which can be proved directly by various combinatorial methods, and hold more generally
for simplicial spheres.
Stanley's theorem yields an explicit numerical characterization of the f  v e c t o r s
o f simplicial dpolytopes, which is expressed in terms o f the hk and the gk (the
gtheorem) [
21
].
T h e o r e m 4 (Stanley). Suppose that h = (h0 . . . . . hd) Z d+l, go = h0, and gk =
hk  hk1, k = 1. . . . . Ld/2J. Then h is the hvector o f a simplicial dpolytope if and
only if'.
1. hi = ddi, i = 0 . . . . . ld/2J.
c. w. Lee
2. gi > 0, i = 0 . . . . . Ld/2J.
3. go  1 and gi+l < giIt? ' t9 ~ 1. . . . . Ld/2J  1.
See, for example, [
21
] for the definition of the pseudopower gy).
2.2.
Kalai's P r o o f
Kalai's proof [
8
] that g2 is nonnegative is quite accessible, but does not have the full
force of the gtheorem. Again, we need to start with some definitions. Let G = (V, E)
be a graph, where V = {1. . . . . n}. Choose a point vi ~ R d for each vertex of the
graph and make a barandjoint structure by placing bars connecting pairs of points
corresponding to the edges of G (we do not worry about selfintersection). Often we
refer to the vi themselves as the vertices, and the bars as the edges. A stress on this
barandjoint structure is an assignment of numbers ~.ij to edges vi vj such that
j: l)ivjEE
)~ij(1)j  Oi) = O
(1)
holds for every vertex /)i. The vector space of all stresses is the stress space of the
structure.
A n infinitesimal motion of the structure is a set of vectors U1. . . . . ~n e R d such that
d([I (vi + t ~ i )  (vj + t ~ j ) 112)/dt = 0 for all edges vi vj. Equivalently, (vi  vj) r (6i ~ j ) =
0 for all edges, or the projections of vi and ~j onto the affine span of {vi, vj } agree. Some
infinitesimal motions are trivial in the sense that they are induced by rigid motions of R d
itself. Motions apart from these are called nontrivial. If the structure admits only trivial
motions, it is infinitesimally rigid.
Using the classical relationship between the space of infinitesimal motions and the
space of stresses of a structure, and the fact that the barandjoint structure associated with
the edgeskeleton of a simplicial convex dpolytope P, d > 3, is infinitesimally rigid
(where we take the vi to be the vertices of P itself), Kalai observes that the dimension
of the stress space of P is g2, and hence g2 must be nonnegative.
In this striking proof of the Lower Bound Theorem Kalai speculates whether it might
be possible to extend the notions of stress and rigidity appropriately to the
higherdimensional faces of P to reprove the nonnegativity of the other gk, and possibly even
find a new proof of the gtheorem. The notion of generalized stress presented below
accomplishes this, but these results depend in an essential way upon McMullen's new
proof of the gtheorem [
13
], [
14
] using weights on simple polytopes.
Generalized Stress
3.1.
Working Toward a Definition
We could define generalized stress by starting with some analog of classical stress or
infinitesimal motion, but instead we work toward the definition by following the path by
which 'it was originally discovered. This route was primarily motivated by attempts to
mimic some aspects o f Kalai's algebraic shifting technique [
7
].
For x = (Xl . . . . . xn), and for (rl . . . . . rn) ~ Zn+, by x r we mean x p ...x~, ~ Define
also s u p p x r = {i: ri ~ 0} (the support o f x r ) , r! = r l !    r n ! , and Irl = rl +    + rn.
Write ei for the vector o f length n consisting of all zeros, except for a one in the ith
position, and e = (1 . . . . . 1).
Let A be a simplicial complex (not necessarily of dimension d  1) With n
vertices {1. . . . . n}, and let R = R[xl . . . . . xn] = Ro (t) R1 (3 RZ (3 "'" be the ring o f
polynomials, graded by degree. Consider any elements 01 . . . . . Od ~ R1. We wish to
determine information about the dimension of Bk (as a vector space over R), where B =
B o ( 3 B I ~ B 2 ~ . . . is the result of taking R and factoring out the ideal J = J o ~ J l ~ J 2 ( 3 . . "
generated by IA and 01. . . . . Oa. Place an inner product on the vector space Rk by defining
( ~ r : Irl=k arxr' Z r : Irl=k brxr} : ~.r:Irl=k arbr. W r i t e gk = Jk (]) J / . It is
straightforward to see that )'~: Irl=k brxr is in J ~ if and only if it is orthogonal to:
1. All monomials of the form xSx q where xq is squarefree, suppxq r A, and
I s l + l q l = k .
2. All polynomials of the form xsOj, where Isl = k  1.
Define vi = (oil . . . . . vial)r, i = 1. . . . . n, where Oj = ~,i"=1 1)iJxi, J = 1. . . . . d. Then
the first condition is equivalent to the condition
b r = 0
if
and the second condition is equivalent to the condition
Define M to be the d x n matrix with columns ol . . . . . on. Then Y]r: IrL=kbrx~ satisfies
condition (3) if and only if
o r
i=1
for every s ~ Z~_ such that Isl = k  1. Thus we have a linear equation on the vectors
1)i for every such s.
The second condition can be expressed more compactly if we look at
where the lefthand side is to be regarded as a polynomial with vector coefficients, or
n
i=1
bs+e, vi = O,
(2)
(3)
(4)
This leads directly to our definition o f generalized linear stress:
D e f i n i t i o n 1. Let A be a simplicial c o m p l e x (not necessarily o f d i m e n s i o n d  1) on
the set {1 . . . . . n}, and let Vl . . . . vn ~ R d. Let M be the d x n m a t r i x with columns
Vl . . . . . on. F o r each k = 0, 1, 2 . . . . . a linear kstress on A (with respect to Vl . . . . . Vn)
is a p o l y n o m i a l o f the form
that satisfies
and
b ( x ) =
~
r: Irl=k
X r
br'7.
br = O
i f
The collection o f all linear kstresses forms a vector space, which is d e n o t e d Ske.(In [
10
]
we used the notation Bk.)
There was evidence to suggest that the Hard Lefschetz element a~ in the p r o o f o f the
g  t h e o r e m could be chosen to be Xl +  . + xn. This was confirmed by M c M u l l e n [
13
],
[
14
]. So we are also interested in the effect o f factoring out xl +    + x , from R as well.
This suggests the definition of g e n e r a l i z e d affine stress:
D e f i n i t i o n 2. Let A be a simplicial c o m p l e x (not necessarily o f d i m e n s i o n d  1) on
the set {1 . . . . . n}, and let Vl . . . . Vn E R d. Let M be the (d + 1) x n matrix obtained
by appending a final row o f ones to the matrix M with columns vl . . . . . vn. For each
k = 0, 1, 2 . . . . . an affine kstress on A (with respect to vl . . . . . v.) is a p o l y n o m i a l o f
the form
o r
The collection o f all affine kstresses forms a vector space, which is d e n o t e d S~. (In [
10
]
we used the notation Ck.)
Equivalently, an affine kstress is a linear kstress that satisfies the additional condition
o r
for every s ~ Z~_ such that Isl = k  1. That is, we have an affine relation on the vectors
vi for every such s.
It is obvious that b ( x ) is an affine kstress with respect to vl . . . . . v~ if and only if it
is a linear kstress with respect to ~1 . . . . . Un, where
~
i=1
bs+e, = 0
, 1
Differential operators with constant coefficients acting on the stress spaces play an
impoltant role. In particular, we can construct an operator that will provide a relationship
between linear and affine stresses, and which is seen in Sections 9 and 10 to serve as the
Lefschetz element in the p r o o f o f the gtheorem.
For c ~ R n, define the function trr on the space of linear stresses by
ar
= c r v b = ~
i=1
n
Ob
ci 8xi
for any linear stress b(x). Define in particular
o) (b) (re (b)
20b
~x/"
T h e o r e m 6. Let A be any simplicial complex with n vertices, and let vl . . . . . Vn E R d.
Then, f o r k = 1, 2, 3 . . . . . the function trc maps S~ into S~e_l, and, f o r k = 0, 1 , 2 . . . . .
the kernel o f to restricted to S~ is S~.
~ a i
i=O o x i
OXo

/ : l O x i
a i
P r o o f
s ( f )
For
and
Conversely, suppose that/~(x0, xl . . . . . xn) ~ S~(A). For a polynomial expression f in
x0 . . . . . xn, define s ( f ) = f(O, xl . . . . . xn). We can check that b = s(/0 is in S~(A):
  S
= S
= 0 .
  a i + ai
a i s
.~ Ob
i=] OXi
~.
v i
~ O .
Why "Stress"?
The use of the terms "linear" and "affine" in the definition has already been
justifiedthe conditions for b(x) to be a stress involve either linear or affine relations on the vi.
However, it is not yet clear why the term "stress" makes sense. This will be motivated
in several stages. First we show that S~ is isomorphic to the classical stress space of
a barandjoint structure. The higher dimensions will require some preliminary work.
However, first, we consider some simple examples.
4.1. Examples
The first example is an easy but important one that will resurface later in this paper.
E x a m p l e 1. Consider a geometric dsimplex in R d and let A be its boundary complex.
Choose Ol. . . . . Vd+l to be the vertices of the simplex itself. Assume further that the
simplex is positioned such that no proper subset of the vertices is linearly dependent.
Then nonzero ci e R exist such that
d+l
and all linear relations on the vi are nonzero scalar multiples o f this one. We claim that
for all k = 0 . . . . . d, S~e is onedimensional and is spanned by
We can verify this by using the fact that
E
r: Irl=k
X r
c r y .
r!
d+l d+l
E Cs+eil)i ~"cs E
i=1 i=1
Cil)i = 0
for all s ~ Za++1 such that Is[ k  1. Observe that cr is nonzero for all r. On the other
hand, di"m S~e = 0 for all k > d, dim S~ = 1, and dim S~ = 0 for all k > 1, since the vi
are affinely independent and so y ' f =+~ c i r
E x a m p l e 2. Suppose that P is the standard octahedron in R 3 with vertices vl =
(1, 0, 0) r, v: = (  1, 0, 0) r, v3 = (0, 1, 0) r, v4 = (0,  1, 0) r, v5 = (0, 0, 1) r, and
V6 ~ (0, 0,  1) r. Then, for the boundary complex A o f P , the stress spaces with respect
to Vl, . . . . v6 are given by:
1. S o e = R .
2. S~ is threedimensional and has a basis {xl + xz, x3 + x4, x5 + x6}.
3. $2e is threedimensional and has a basis {(xl + x2)(x3 + x4), (xl + x2)(x5 + x6),
(X3 '[ X4)(X 5 '[ X6) }.
. $3e is onedimensional and has a basis {(xl + xe)(x3 + x4)(x5 + x6)}.
5. Se = {0} i f k > 3.
6. S~=R.
7. S~ is twodimensional and has a basis {xl + x2  x3  x4, xl + x2  x5  x6}.
8. St = { 0 } i f k > 1.
4.2.
Connection with Classical Stress
Turning now to general simplicial complexes, we can describe the lowdimensional stress
spaces and clarify the connection with classical stress:
T h e o r e m 8. Let A be any simplicial complex with n vertices, and let vl . . . . . vn E R a.
Then:
1. S ~     S ~ = R .
2. Sel is isomorphic to the space o f all linear relations on the vectors vl . . . . . vn.
3. S t is isomorphic to the space o f all affine relations on the vectors Vl . . . . . on.
4. S~ is isomorphic to the classical stress space on the barandjoint structure where
the vertices are placed at the points vl . . . . . On, under the correspondence )~ij =
bei + bej f o r all i ~ j .
EL.Spheres, Convex Polytopes, and Stress
Proof. Only the fourth part requires any explanation. Assume b ~ S~. Set )~ij = be,+ej
for all i, j = 1. . . . . n. Of c o u r s e , )~ij ~ )~ji, and )~ij = 0 if {i, j} is not an edge of A.
From conditions (3) and (5) we find that, for all j = 1. . . . . n,
i: i # j
)~ij 1)i "at)~jj 1)j
i: i~j
~ . i j ( o i  vj),
where E is the set of edges of A. Therefore the ~'ij s a t i s f y the equilibrium condition (1).
Conversely, assume we have numbers ~.ij for each {i, j } ~ E that satisfy condition (1).
For j = 1. . . . . n define
and for i # j define
bjj =
~
i: {i.j}~E
)~ij,
bei+eJ : {oi j
ioftherwise.{i'j}eE'
Reversing the previous calculations shows that the resulting quadratic polynomial b ( x )
is an affine 2stress. []
4.3.
Coefficients o f SquareFree Terms
Our next step is to show that under suitable conditions the coefficients of the squarefree
monomials of a linear or affine kstress uniquely determine the remaining coefficients of
the polynomial. We then concentrate our attention on the squarefree terms, regarding
the coefficients as assignments of numbers to various faces of the simplicial complex,
and give a geometric necessary condition on these numbers that turns out to be a natural
generalization of classical stress. We are, in fact, able to give explicit formulas for the
coefficients of the nonsquarefree monomials in terms of the coefficients of the
squarefree monomials, and in the process show that the above necessary condition is also
sufficient and thus characterizes the coefficients of the squarefree terms.
For a simplicial complex A with n vertices and for Vl. . . . . v, ~ R d, we say that the vi
are in linearly general position with respect to A if {vii . . . . . vi, } is linearly independent
for every face {il . . . . . is} of A.
T h e o r e m 9. Let A be any simplicial comptex with n vertices and assume that vl . . . . . v, are in linearly general position with respect to A. I f b(x) is a linear stress, then the coefficients o f the nonsquarefree monomials in b(x) are linear combinations o f the coefficients o f the squarefree monomials and hence are uniquely determined by them.
Proof. Let b ( x ) ~ S[. We use reverse induction on q = c a r d ( s u p p x r ) . The result is
trivially true if q = k, so assume the result is true for some q such that 2 < q < k and
suppose that c a r d ( s u p p x ~) = q  1. Choose j such that rj > 1 and let s = r  ej.
Condition (3) implies
n
E
i=1
bs+e~vi = O.
However, by the induction hypothesis the coefficients bs+e, are linear combinations o f
the coefficients o f the squarefree m o n o m i a l s when ri = 0, since c a r d ( s u p p x s+ei) = q in
this case. This leaves the q  1 coefficients bs+e, for i 6 supp x r to be uniquely d e t e r m i n e d
since the corresponding vi are linearly independent by assumption. In particular, b~+ej =
br is a linear combination o f the coefficients o f the squarefree m o n o m i a l s . []
The above p r o o f shows how conditions (2) and (3) can be used in a systematic w a y
to find all the coefficients o f b(x) if the coefficients o f the squarefree terms are given.
C o r o l l a r y 2. Let A be any simplicial complex with n vertices and let vl . . . . . vn E R a
be chosen in a linearly general position with respect to A. Then dim S e = 0 f o r all
k > d i m A + l .
P r o o f In the case that k > dim A + 1 there are no faces o f cardinality k, so all
coefficients o f squarefree m o n o m i a l s o f a linear kstress must be zero. []
4.4. A Geometrical Interpretation o f Stress
For F = {ii . . . . . is} E A, define conv F (with respect to vl . . . . . vn) to be
conv{vfi . . . . . or}. In an analogous way, define a f f F and span F . We sometimes abuse
notation and write bF and x F for br and x r, respectively, where ri  1 if i 6 F and
ri = 0 i f / ~ F . We also use the notation F + i for F U {i} and F  i for F \ { i } . Finally,
if i 6 F , by bF+i we mean br+ei, where r is as above.
T h e o r e m 10. L e t A b e a n y s i m p l i c i a l ( d  1 )  c o m p l e x w i t h n v e r t i c e s a n d l e t v i . . . . . vn
R a. Let b ( x ) be a linear (resp. affine) kstress, k > 1. Choose any f a c e F o f A o f
cardinality k  1 and any point v in span F (resp. aft F). Then
lies in span F (resp. aft F). Equivalently, if wi is the vector joining the projection o f vi
onto span F (resp. aft F ) to vi, then
bF+i(1)i  1))
i ~lk F
bF+iWi ~ O.
(6)
Proof.
Suppose that v E span F. Then, using condition (3),
bF+i(1)i  U) :
1)'I y ~ . b F + i V i  ~
i~lk F i~lk F
bF+iV
~ 1 )   E b F + i l ~ i 
iEF
E
iElk F
bF+il)
which is in span F. I f b is an affine stress, then by condition (5) the sum o f the coefficients
in the above expression is
E
iElkF
b F + i
bF+i = 1.
So we have an element o f aft F.
Note that for linear kstress, w i is the altitude vector for the point vi in the simplex
conv({O} U ( F + i)), and for affine kstress, wi is the altitude vector for the point vi
in the simplex c o n v ( F + i). In particular, condition (6) for affine 2stress is identical to
condition (1) defining classical stress. So affine kstress generalizes classical stress in a
natural way, and could in fact have been defined by condition (6) in the first place. This
is the definition that Kalai was thinking of (personal communication). Linear kstress
seems less natural at first sight since it is dependent upon choice o f origin. In the case o f
simplicial polytopes, however, we see in Section 10 that linear stress becomes invariant
under rigid motions when dualized and interpreted as McMullen's weights on simple
polytopes.
E x a m p l e 3. Let P be a simplicial convex dpolytope in R d, A its boundary complex,
and Vl . . . . . Vn its vertices. Then the above theorem shows that dim S~ = 0. Take any
b ( x ) E S~ and consider any subfacet F (i.e., o f cardinality d  1). There are exactly two
facets containing F and hence only two altitude vectors wi with respect to aff F , where
i E lk F. By convexity these two vectors are not collinear and we know
~~ bF+iWi ~ O,
iElk F
from which it follows that bF+i ~ 0 for i E l k F. Thus all the coefficients o f the
squarefree monomials o f b ( x ) are zero, and so all o f the remaining coefficients must likewise
be zero.
4.5.
Formulas f o r the Coefficients
Condition (6) is a nice geometrical necessary condition for the coefficients o f the
squarefree terms of generalized stress. However (again with suitably general vi), this condition
is also sufficient, as we now show.
Assume A is a simplicial complex of dimension at most d  1 with vertices 1. . . . . n,
and that /)1 ..... On E R d are in linearly general position with respect to A, Assume
further that Ul . . . . . Ud E R a.
Suppose that G = {il . . . . . is} _c {1 . . . . . n}, where s < d. Fix an ordering o f the
elements o f G and define
Theorem 11. Let A be a simplicial complex on n vertices o f dimension at m o s t d  1,
let vl . . . . . vn be in linearly general position with respect to A , and let Ul . . . . . Ud be
in linearly general position with respect to A and vl . . . . . on. Suppose that we have
numbers be assigned to each (k  1)face F o f A that satisfy condition (6). For each
r 9 Zn+ such that ]r[ = k and S = s u p p x r 9 A, define
br =
Y ~
(k  1)facesF containingS
bF ~ I [ F  i]ri1..
i~F
Then b ( x ) = E r : ]rl=k b r ( x r / r .!) is a linear kstress.
Proof. We already know that there can be at most one linear kstress b ( x ) with the
given coefficients o f the squarefree terms. We must show that in fact there is one, and
that it is given by the formula above. Consider one instance o f condition (3):
n
y ~ bs+ejVj = O,
j = l
where [s[ = k  1. Let S = s u p p x s. The coefficients bs+ej appearing in the
expression correspond to monomials with support size either c a r d ( s u p p x s) (if j E S) or
card(suppx s) + 1 (if j 9 lk S). So we can contemplate the possibility o f using these
conditions repeatedly to determine the coefficients o f monomials with smaller supports
from the coefficients o f monomials with larger supports. In the process we need to verify
that:
(i) For a given instance of the condition it is possible to solve for the unknown
coefficients, i.e., that
j elk S
bs+ejVj 9 span{vj: j 9 S}.
(7)
(ii) I f the same coefficient is determined from two different applications o f
condition (3) in this manner, that we do not get contradictory values.
The p r o o f will therefore be by reverse induction on p = card(suppxr). The formula
stated in the theorem is trivially true if k = 1 or if p = k so we assume that k > 2 and
1 < p < k. Choose any m for which rm > 1. Let s r  e m and S = supp x s = supp x r.
EL:Spheres,ConvexPolytopes,and Stress
tl
E
i=1
bs(j)+eil3i = O.
Wedge this with Vsq+j and sum over all j r S:
Z
jr
~
i=1
(bs(j)+eiVi A 1)s_q+j)= O.
If i = j or if i ~ S  q, then vi A Vs_q+j ~ O. If i r S, then one of the terms in the
above expression is
However, interchanging the roles of i and j also yields the term
These terms cancel since s ( j ) + ei = s(i) + ej. Looking at the remaining terms (where
i = q) we see
~"~(bs(j)+eq 1)q A 1)S_q+j) = O.
yes
However, s ( j ) + eq = r  eq + ej + eq = r + ej, so
bs(j)+e~ vi A vj A VSq.
bs(i)+ej vj A 73i A 1)S_q .
0 = ~_,(br+e, vj A VS)
jCS
Therefore
Cramer's rule can be used to solve for br (which equals bs+ej when j = m) in the system
Z
j~s
bs+e, V j =   E b s + e , Vj,
j~lkS
d
E b s + e j l 3 j 't Z
jES j=p+l
c j u j =
giving
br
IS  m + (Y~7~tt s bs+ejvj ) ]
is]
Applying induction and using some GrassmanPlticker relations,
The fact that this final formula is independent o f the choice o f m shows that we will not
get contradictory values for br from different choices o f s. []
Filliman [
4
] and Tay et al. [
22
] have also shown the sufficiency o f condition (6), but
without the explicit formula above. (Unfortunately, however, the proof o f the g  t h e o r e m
for p.1.spheres in [
4
] is incorrect.)
The relationship between the space of classical stresses (affine 2stress) and the space
o f classical infinitesimal motions o f barandjoint structures is straightforward: they are
the left and right nullspaces of a c o m m o n matrix. This suggests a natural way to define
generalized infinitesimal motions associated with affine kstress.
Suppose that b(x) is an affine kstress o f a simplicial complex A with respect to
vl . . . . . Vn ~ R d. Looking at the equivalent condition on coefficients o f squarefree
terms, we have a number bF assigned to each (k  1)face F , such that, for every
(k  2)face G,
~~ bG+iWi ~ O,
i~lkG
EL:Spheres, Convex Polytopes, and Stress
where wi is the altitude vector o f ~)i in the simplex conv(G + i). We can express these
conditions in terms o f a matrix R whose rows are indexed by the (k  1)faces F and
whose columns occur in groups of d with one group for each (k  2)face G. In the
row corresponding to F and the group o f columns corresponding to G we place the row
vector o f length d:
O r
w (rG,F)
if
if
G ~ F,
G C F.
Here, W(G,V) is the altitude vector o f the simplex cony F with respect to conv G. Then the
left nullspace o f R is S~. A n infinitesimal kmotion will then be defined to be an element
o f the right nullspace o f R, and is described by an assignment of a vector ~ c E R d to
each (k  2)face G such that, for every (k  1)face F ,
We have condensed the notation, writing wi for W~Fi,F) and 6i for v F  i .
We can reformulate this condition by writing Gi for F  i and ui for the unit outer
normal vector o f conv Gi with respect to conv F in aff F. This yields
~
iEF
w i .vi = 0 .
~__j Ui Uwi II 9~i = 0,
iEF
2 _ , ui 9 v o l k  2 ( a i )  O.
iEF
,7 .VOlk2(Gi)(Ui "mi) = O,
iEF
which upon dividing by volk_l ( F ) implies
Therefore
where mi = ~i/vol2_2(Gi).
So an infinitesimal kmotion is a choice of vector ~ G E R d for each (k  2)face G
such that
GcF
vOlk2(G)(uG 9~ )
for every (k  1)face F , where u 6 is unit outer normal of conv G with respect to conv F
in a f f F .
T h e o r e m 12. Suppose that vectors ~ G are given f o r each (k  2)face G. Then the
following two conditions are equivalent:
1. The ~ G constitute an infinitesimal kmotion.
2. For each (k  1)face F a vector mF E R a parallel to F exists such that'mF "UG =
mG " U6 f o r all G C F.
One direction is easy. Suppose that (2) holds. Then, for each (k  1)face F,
E
GcF
volk2(G)(UG" raG) = E
GcF
VOlk2(G)(UGmF)
= O . m F
by Minkowski's theorem. So (1) holds.
On the other hand, suppose that (1) holds. Fix F and regard the UG and the m 6 as
sitting naturally in R k1 and constructthe (k  1) x k matrix U whose columns are the
u6. The rows of U are linearly independent, so the left nullspace of U has dimension
one. Now the vector y whose entries are the volk_2(G) is in the nullspace of U, and the
vector z whose entries are the UG 9~ G is orthogonal to y. Therefore z is in the rowspace
of U, and in particular a single vector ~F exists such that ~ e 9U = z. []
This gives us equivalent formulations of condition (8). Suppose that vectors ~ are
given for each (k  2)face G. For a (k  1)face F and G C F, let m e be the projection
o f m G onto the (k  1)dimensional linear space V parallel to F. For a real number t let
F ( t ) be the (k  1)simplex determined by translating a f f G by the vector t m a .
Corollary 3. The following conditions are each equivalent to condition (8):
1. Y~GcF volk2(G)(u6 9m a ) = O.
2. F ( t ) is congruent to F.
3. F(1) is congruent to F.
4. ( d / d t ) vol2_l ( F ( t ) ) = O.
Proof. (1) is clear since uG 9m a = u6 9m e , (2) and (3) hold since by the theorem
we are equivalently translating each a f f G by the same vector mF. (4) then follows
immediately. []
The last condition is a very natural generalization of the definition of classical
infinitesimal motion (infinitesimal 2motion) and was also observed by Filliman [
4
]. See
[
22
] for a deeper study of the relationship between generalized stress and skeletal rigidity
of cell complexes (not necessarily simplicial).
6. Bistellar Operations
In this section we examine how the various stress spaces change under the action of certain
local changes in a simplicial (d  1)complex A. Let F and G be disjoint nonempty
subsets of {1. . . . . n} of cardinality p and q, respectively, such that p + q = d + 1,
F E A, G ~ A, and lk F = OG = {G': G ' is a proper subset of G}, the boundary of G.
ELSpheres, ConvexPolytopes,and Stress
The simplicial complex A' = ( A k F ) tO (G 9 OF) is the result of performing a bistellar
operation on A. During this operation we remove all faces containing F and introduce
all sets of the form G to F ' where F ' E OF. The faces of A' that are new are those faces
of A' that contain G, and the faces of A which are lost are the faces of A that contain F.
The local change in the structure of A induces a corresponding simple "local" change
in the linear stress spaces.
T h e o r e m 13. Assume A and A ' are as above and that vl . . . . . vn are in linearly general
position with respect to both A and A'. Then
Proof. The main idea is to use an intermediate simplicial complex A" to mediate the
changes in the stress spaces. Define A" = A tO ( G . OF) and observe that A" also equals
A' tO ( F . OG). We will show that
[ d i m Sse(A)
dim S*e(A") = / c t i m s [ ( z x ) + 1
if O < s < q  1 ,
if q < s < d ,
and by symmetry
] dim Sst (A')
dim S~e(A") = / d i m S~e(A') + 1
if 0 < s < p  1 ,
if p < s < _ d .
Since A and A" share the same faces of cardinality s when 0 _< s _< q  1, then Sse(A)
must be the same as Ses(A ") for these values of s. Assume that q _< s < d and take
S = F D G, a subset of cardinality d + 1. All of the proper faces of S are in A". Define
c ( x ) to he the unique (up to scalar multiple) linear sstress on the simplicial complex
consisting of all subsets of S as constructed in Example 1. Each face of A is also a face
of A", so S,e(A) _ SaC(A"). Suppose that b ( x ) is a linear sstress that is in A" but not
in A. This implies that br is nonzero for some r such that suppx r E openstar G (the
set of all faces of A" that contain G). We claim that the restriction of b to the faces of
S must be a nonzero multiple of c(x); i.e., that there is a nonzero real number t such
that br = tCr for all x r supported on openstar G, and hence b ( x )  t c ( x ) is in Sse(A).
Since G is the only face of cardinality q in openstar G, this is clearly true if s = q. So
assume q + 1 < s < d. Choose any r such that br is nonzero and suppx r E openstar G.
Since s > q, there is a j such that supp x rej ~ openstar G. Condition (3) implies that
E i n= I b rej+ei 1)i ~" O. AS this sum involves only the d + 1 vectors vi such that i E S, the
coefficients in this sum must be a common multiple of the corresponding coefficients
of c(x). Using this procedure repeatedly to determine the other coefficients br verifies
the claim. The resulting direct sum decomposition of Sse (A") establishes the change in
dimension. []
7.1. P.L.Spheres
7. Simplicial Spheres, ELSpheres, and Pseudomanifolds
Bistellar operations are ideally suited for proofs by induction on p.1.manifolds, especially
considering Pachner's result [
17
] that any two p.1.manifolds that are p.l.homeomorphic
can be transformed into each other by a sequence of bistellar operations. In particular,
every p.1.sphere can be obtained from the boundary of a simplex by such operations,
and the discussion in the previous section almost immediately proves:
Corollary 4. I f A is a simplicial p.l.sphere, then A is CohenMacaulay.
Proof Assume that A is a (d  1)dimensional simplicial p.l.sphere with vertices
{1. . . . . n}. Choose vl . . . . . vn E R d in linearly general position with respect to all
subsets of {1. . . . . n} of cardinality d. The boundary of a (d  1)simplex is C o h e n
Macaulay by part (1) of Corollary 1 and Example 1, since its hvector equals (1 . . . . . 1).
It is well known that the components of the hvector of a simplicial complex change
under the action of a bistellar operation in exactly the same way as the changes in the
dimensions of the linear stress spaces described in Theorem 13. So since A can be
obtained from the boundary of a simplex by a sequence of bistellar operations, we have
hs(A) = dim S[ for all s. Therefore A is CohenMacaulay by Corollary 1. []
7.2. Pseudomanifolds
We now turn to a larger class of simplicial complexes which includes simplicial
manifolds. A simplicial (d  1)complex is said to be a pseudomanifold if:
(i) Every maximal face has dimension d  1.
(ii) Every (d  2)face is contained in exactly two faces of dimension d  1.
(iii) Any two (d  1)faces can be connected by a path of (d  1)faces, each two
succeeding faces of which are adjacent (share a common (d  2)face).
T h e o r e m 14. I f A is an orientable ( d  1)pseudomanifold on n vertices and vl . . . . . vn
are in a linearly general position with respect to A, then dim S~(A) = 1.
Actually, Tay et al. [
23
] prove the stronger result that the dimension of Sde(A) equals
the dimension of the homology Hd ( A , R), and the proof of the above theorem hints why
this is true.
Proof Let b(x) be a linear dstress on A. By Theorem 9 it suffices to study the
squarefree coefficients of b(x). Choose a consistent orientation of all the facets ((d  1)faces)
of A and use this to induce an ordering of the elements of each facet. Let G be a subfacet
((d  2)face) of A, and let F1 and F2 be the two facets containing G. Theorem 10
implies that [F1]bEF,~ = [Fz]bEF+ so a constant t exists such that bF t[F] 1 for every
EL.Spheres, ConvexPolytopes, and Stress
facet F. The coefficients of the nonsquarefree terms are then uniquely determined. So,
up to scalar multiple, there is only one element in Se(A). []
In Section 9 the geometrical significance of this canonical linear dstress (the one
for which be = [F] 1 for each facet), in the case that A is the boundary complex of a
simplicial convex polytope, will b e c o m e apparent.
Suppose that A is a simplicial (d  1)complex on {1 . . . . . n} and Vl . . . . . vn ~ R a.
For G = {il . . . . . is }, a subset of {1. . . . . n}, define the function r e on the space of linear
stresses by
In particular, write
z t ( b ) =
aSb
Oxi~ . . . Oxi, "
vi (b) . . . .
Ob
Oxi
.
T h e o r e m 15. Let A be a simplicial orientable (d  1)pseudomanifold on {1 . . . . . n}
and let vl . . . . . vn E R d be in linearly general position with respect to A . Suppose that
b ( x ) is the canonical linear dstress o f A and G is a f a c e o f A o f cardinality s. Then
rG(b) is a linear (d  s)stress supported on clstar G. In the special case that l k G is
a (d  s  1)sphere, then up to scalar multiple rG(b) is the unique nonzero linear
(d  s)stress supported on clstar G.
Proof. The first part of the theorem is obvious: if the coefficient of x r is nonzero in
rG (b), then the coefficient of Xil " . . x i , x r must be nonzero in b (x). Hence G U (supp x r)
is a face of A and so s u p p x r ~ clstar G.
For the second part, note that clstar G can be obtained by starting with Ik G and
successively joining it to the vertices of G. Since lk G is a (d  s  Dsphere, it has a
unique linear ( d  s)stress (up to scalar multiple). By repeated application of T h e o r e m 7,
so does clstar G. Now it is easy to see that z~(b) is nonzero since b ( x ) is, and so z t ( b )
must be a generator of the linear (d  s)stresses on clstar G. []
8. S h d l i n g s
Consider a simplicial (d  1)complex on {1, . . . , n} such that every maximal face has
dimension d  1 (is a facet). Then the complex is said to be shellable if the facets can
be ordered F1 . . . . . Fm in such a way that, for k = 1. . . . . m, there is a unique minimal
face Gk that is in F~ but is not in Uik__~~ i . Here, F i denotes the simplicial complex
consisting of all subsets of F,. It is well known that, as each facet Fk is added, precisely
one component hs of the hvector increases by one, the remaining components being
unchanged; specifically, s = card Gk. Using this and understanding the changes in the
face ring during the shelling, Kind and Kteinschmidt [
9
] give an inductive proof that
shellable simplicial complexes are CohenMacaulay.
It is also possible to use generalized stress to prove this result by showing that the
dimension of Sse increases by one when Fk is added, while the dimensions of the other
linear stress spaces do not change. However, we content ourselves with considering the
special case when A is a simplicial (d  1)sphere. Assume that vl . . . . . Vn ~ R d are in
linearly general position with respect to A, and let b ( x ) be the canonical linear dstress
on A. When Fk is added, the closed star of G~ = F k \ G k is completed. If card Gk = s,
then card G~ = d  s, and Theorem 15 implies that up to scalar multiple there is a unique
linear sstress rG; (b) supported on clstar G~. The coefficient of this stress associated with
the face Gk is nonzero, so this stress was not present before Fk was added. So we can
use the shelling of A to derive a basis for the stress spaces.
T h e o r e m 16. l f A is a sheUable simplicial (d   1)sphere whose n vertices, vl . . . . . vn 9
R d, are in linearly generalposition with respect to A , and F1 . . . . . Fro, G] . . . . . G~ and
b(x ) are as above, then {rr, (b): card G'k = d  s} is a basis f o r Ses. Hence the collection
{rG(b): G is a f a c e o f A o f cardinality d  s} spans Se~.
9. Simplicial Convex Polytopes
In this section we specialize further and consider the case when A is the boundary
complex of some simplicial convex dpolytope P C R a. This was the motivating case
for defining generalized stress in the first place and trying to understand the gtheorem.
9.1.
Canonical Stress and Volume
Assume that P contains the origin in its interior. Then the vertices vl . . . . . on of P are in
linearly general position with respect to A. Since A is shellable, we know dim Se = hi,
i = 0 . . . . . d.
The definition of affine stress seems more geometrically natural for simplicial
complexes since affine stress is invariant under translation. The linear stress spaces, while
also geometrically definable, depend upon the choice of origin and change with
translation. It turns out, however, that this situation changes entirely when we turn to the polar
P* of P and describe the linear stresses in terms of conditions on P*. This will become
clearer in Section 10, but already in this section we begin to see the significance of using
the polar to understand stress.
F o r x 9 Rn, consider the polytope Q ( x ) = {y 9 Rd: y T v i < Xi, i = 1. . . . . n}. O f
course, Q(e) is the polar P* of P. Since P* is simple, for values o f x i n e a r 1, Q ( x ) and
P* are strongly isomorphic. It is well known that the volume of Q ( x ) as a function of
the xi is a homogeneous polynomial V ( x ) = )~r:Irl=db r ( x r / r ! ) of degree d, br = 0
whenever suppx r r P, and bF ~ [ F ] 1 for every facet F of P. The canonical linear
dstress b(x) on A also shares these properties, and so perhaps the following result is
not completely unexpected:
Theorem 17. Let P be as above. Then the canonical linear dstress is precisely V (x ).
F o r e v e r y u 9 R d, Q ( x I . . . . . Xn) t u = Q ( x 1 IuT1)I . . . . . Xn + uTvn) ( w e a r e
O =
[ V ( X l . . . . . Xn)  V ( x 1 + uTo1 . . . . . X n + uTpn)]
~~ br+eiUT tli
i=1
u T
\ i = l
br+e Vi 9
' /
However, this is true for every u, so Y~7=1br+e, Pi = 0 and V(x) is a linear dstress.
That V(x) is the same as the canonical linear dstress follows from the fact that the
coefficients of the squarefree terms of V ( x ) agree with those o f the canonical linear
dstress. []
9.2. LowerDimensional Canonical Stresses
The above proof mimics the proof o f Minkowski's theorem that
.n~vob_~(~)/)~i = o ,
which we already used in Section 5. (F1 . . . . . Fn are the facets o f P* corresponding to
the vertices Vl. . . . . vn of P.) In fact, the relationship between Minkowski's theorem and
stress is quite strong, as we will see.
One way to prove the gtheorem would be to show that the application o f 09d2i
induces a bijection between Sed_iand S/e, i = 0 . . . . . [ d / 2 J . Actually, it would suffice to
show that 09: S/e + Se_l is surjective for i = 1. . . . . Ld/2J. McMullen's new proof o f
the gtheorem shows that the bijections proposed here are valid.
Given the canonical linear dstress V (x), we might consider applying 09 repeatedly
to get canonical linear istresses 09di(V(x)), i 0 . . . . . d  1. Let W ( x ) = V(xl +
1. . . . . xn + 1). Then, for small x, W ( x ) is the volume o f a polytope near P*. Write
W(x) = )]/d=l Wi(x), where Wi(x) is a homogeneous polynomial o f degree i, i =
0 . . . . . d. It is clear that the constant Wo(x) is the volume o f P* and Wd(X) = V(x). It
is also easy to see that W1(x) = )]~=1(VOld1 (Fi)/ll vi II)xi.
T h e o r e m 18. Let P be as above. Then J  i ( V (x)) = (d  i)! Wi(x), i = 0 . . . . . d.
Proof. We calculate the contribution o f br(xr/r!) in V ( x ) to the coefficient o f x s in
Wi (x), where x s Ixr. Expanding
br (Xl + 1) r~ " ' " (xn + 1) r"
r l ! . . 9rn!
we see that the contribution is
On the other hand, the contribution of
in V ( x ) to the coefficient o f x s in w d  i ( V ( x ) ) , where i = d  [s[, is
[]
Corollary 5. Let P be as above.
1. The canonical linear Ostress ofl (V (x )) equals d! vol(P*).
2. The canonical linear 1stress w d1 (V (x )) equals
(d  1)! E
i=l
?1
VOldl(Fi)
[11)i[I
Xi"
That is, the canonical linear combination of the vi induced by co is (up to scalar multiple) the same as that induced by Minkowski' s theorem.
We can find the coefficient of the squarefree term of Wi corresponding to an (i 
1)face F of P by looking at the corresponding (d  / )  f a c e F* of P* and computing the
contribution to the local change in the volume of P* due to the translations of the facets
containing F*. This change depends upon the (d  / )  v o l u m e of F* and the size of the
cone of the associated normal vectors, rescaled to account for the fact that they may not
be of unit length.
Theorem 19. The coefficient of the squarefree term of Wi corresponding to F is
VOld_/ (F*)
voli(conv({O} kJ {vi: i E F}))"
See also [
4
].
So each Wi is associated in a very natural way with the (d  / )  v o l u m e s of the
(d  / )  f a c e s of P*.
Notice that we can write
d
W(x) = ~
i=0
Wi(x)
= Z
i=O
= Z i=0
d t o d  i ( v ( x ) )
(di)T
d w i ( V ( x ) )
i!
P.L.Spheres,Convex Polytopes, and Stress
9.3.
Easy Bijections
Some of the bijections associated with 09 can now be confirmed.
T h e o r e m 20, Let P be as above. Then Cod: SJ ~ S~ is a bijection. Further, if d > 3, then wd2: Se_l ~ Sel is a bijection.
Proof The first statement is trivially true by part (1) o f the previous corollary
simply because P* has positive (and hence nonzero) volume. From the D e h n
Sommerville relations we know that dim Sde_l = hdi = hi = dim S(. So it suffices
to show that 09d2:S~_1 ~ S~ is a surjection. F r o m T h e o r e m 16 we know that
{ r l ( V ( x ) ) . . . . . rn(V(x))} spans SJ_ 1. We need to show {09d2r1(V(x)) . . . . .
wd2rn (V (x))} spans S~. Since dim S~ = h l = n  d, it is sufficient to demonstrate that
the given subset of S~ has rank n  d . However, since 09and ri commute, this subset equals
( d  2 ) ! {rl (W2 (x)) . . . . . r, (W2 (x)) }. It is straightforward to check that )~__1xi ri (W2 (x))
= 2W2(x). It is known from the B r u n n  M i n k o w s k i theory that the quadratic f o r m W2(x)
has d zero eigenvalues (associated with the space of translations x = (u r Vl . . . . . u r v,)),
one positive eigenvalue, and n  d  1 negative eigenvaiues; see, for example, [
5
]. So
the rank of the quadratic form is n  d, as is required. Note that this is the case r = 1 of
the H o d g e  R i e m m a n n  M i n k o w s k i inequalities developed by McMullen [
13
]. []
This theorem implies that 09: Se ~ S( is a surjection when d > 3. Therefore
dim S~ = h2  hi = g2 > 0, and we have bound together the main ideas of Stanley's
and Kalai's different proofs of the Lower Bound Theorem.
9.4. Simplicial 3Polytopes
What we have done so far essentially gives a complete description of the situation for
simplicial 3polytopes. If P above is threedimensional, then:
1. The canonical linear 0stress is 3 ! times the volume of P*.
2. The canonical linear 1stress is equivalent to the linear relation induced by
Minkowski's theorem.
3. The canonical linear 2stress is the classical Maxwell stress (shown by
FiUiman [
5
]).
41 S e = R .
5. Sf has dimension n  3 and is isomorphic to the space of all linear relations on
the vi.
6. $2e has dimension n  3, is spanned by the r i V ( x ) , and is isomorphic to S~ under
the bijection induced by multiplication by 09.
7. $3e is s p a n n e d b y V ( X ) .
8. S ~ = R .
9. S~ has dimension n  4 and is isomorphic to the space of all affine relations on
the vi.
10. S~ and S~ are trivial.
11. That o93:$3e + Sot is a bijection is equivalent to P* having nonzero volume.
12. That o9:$2e ~ S~ is a bijection is equivalent to infinitesimal rigidity of the
edgeskeleton of P and and is a consequence of the BrunnMinkowski theory.
So already in dimension three there is a striking confluence ofgeometric and algebraic
results.
Relationship to Weights
10.1. Ring o f Differential Operators
Let A be the boundary complex of a simplicial convex dpolytope P containing
the origin in its interior, and let Vl. . . . . vn be the vertices of P. Consider the ring
R[O/OXl . . . . . O/OXn] of all differential operators with constant coefficients in the
variables xl . . . . . xn. Define the polynomial V ( x ) as before and factor out the ideal of
operators that annihilate the polynomial V ( x ) . Khovanskii (personal communication)
observes that the resulting ring D is isomorphic to the cohomology ring of the projective
toric variety associated with P. This implies the following result, which can be proved
directly.
Theorem 21. Let A be as above. Then D is isomorphic to B = A/(01 . . . . . Od) where
the coefficients o f the Oi are related to the coefficients o f the vertices o f P as in Section 3.2.
Proof. Clearly, r s ( V ( x ) ) equals zero for any subset S r A. However, the invariance
of the polynomial V (x) under translation (see the proof of Theorem 17) implies that
a v ( x )
i=l 1)ij OXZ
_ 0
for each j = 1. . . . . d. Finally, Theorem 16 implies that the image of V ( x ) under
the homogeneous differential operators of degree k spans Sde_~,hence has dimension
hdk hk. Using Theorem 2, this suffices to prove that D is isomorphic to B under the
map O/Oxi ~ X i . []
This viewpoint allows us to define a multiplication on stresses. Let a ( x ) and b ( x ) be
linear stresses. Find operators a' and b' such that a' ( V (x ) ) = a (x ) and b' ( V (x ) ) = b(x ).
Define a ( x ) . b(x) to be the linear stress ( a ' b ' ) ( V ( x ) ) . The multiplication is well defined,
f o r i f a " ( V ( x ) ) = a ( x ) and b " ( V ( x ) ) = b(x), then
(a"b"  a'b') (V (x)) = (a"b"  a'b" § a'b"  a'b') (V (x))
= b"(a"  a ' ) ( V ( x ) ) + a'(b"  b ' ) ( V ( x ) )
~  0 .
It is also clear that the product o f a (d  / )  s t r e s s and a (d  j )  s t r e s s is a (d  i  j )
stress. Writing ~/e = S~_i" we then regard the space o f all linear stresses as a graded
algebra S0e <9 ~ Sae.This algebra is isomorphic to D, hence also to B.
It sometimes helps to take a slightly schizophrenic viewpoint, on the one hand thinking
o f a linear stress a(x) as a polynomial, and on the other identifying it with the operator
a ' for which a'(V (x)) = a(x).
10.2.
Weights
McMullen reproved the gtheorem using the notion of weights on polytopes. An iweight
on a convex polytope P is a realvalued function a on t h e /  f a c e s of P which satisfies
the Minkowski relation
Z a(F)UF, G = 0
FcG
for each (i + Dface G o f P. Here the sum is taken over all ifaces F contained in G,
and UF,G is the unit outer normal vector o f F with respect to G within a f f G . Clearly,
one n a t u r a l /  w e i g h t is given b y a ( F ) = v o l i ( F ) for each iface F. We call this the
canonical iweight. The real vector space o f /  w e i g h t s on P is denoted ~ i ( P ) , and we
denote ( ~ = o ~ i ( P ) by ~ ( P ) . McMullen [
14
] defines a multiplication on f ~ ( P ) that
endows ~ ( P ) with a graded algebra structure.
T h e o r e m 22. Let A be the boundary complex of a simplicial convex dpolytope P C
R d containing the origin in its interior, and take Vl . . . . . vn to be the vertices of P. Then
S/e(A) is isomorphic to f2ei( P*) as vector spaces.
Proof. Suppose that b(x) is a linear kstress. Then, by condition (6), for every face G
o f cardinality k  1 we have the condition
Z
i~lk G
bG+iWi = O.
For a particular i ~ lk G and F  G + i, wi is in the same direction as the corresponding
uG*,r*. So, writing (G) for vold_k(conv({O} U {vi: i ~ G})) and similarly for (F), we
have
E
F=G+iDG
~~ bG+itOi :
iElkG
O,
bFUG,.F*IIWill = O,
E
F~G
bFUG*'F* (F) = O,
(G)
E
F*cG*
bFUG%F,(F) = O.
Hence taking a(F*) = b F ( F ) for each (d  k)face F* yields a (d  k)weight
on P*. []
Although we do not give the details here, it can be shown that the above map is an
algebra isomorphism from S0~ ~ .   ~ Sde tO f2(P*), and that the ring D of differential
operators is isomorphic to the polytope subalgebra FI (P*) defined by McMullen [
13
].
a pi
Let p = ~, and Pi = Wi, which corresponds to the canonical/weight on P*. Then
Pi = pi / i ! (keeping in mind our willingness to confuse at our convenience a differential
operator with the image of V(x) under the action of the operator). Formally writing
[P*] = ~_,di=lPi, we have
This corresponds to the result of McMullen [
13
], [
14
] that [P*] = e x p p and p =
log[P*].
10.4.
Restrictions
The dual interpretation of Theorem 15 is interesting. Let G be a face of P of cardinality
s, and let a(x) be the unique (up to scalar multiple) (d  s)stress supported on clstar G.
Then a(x) is dual to the necessarily unique sweight a* supported on the set of all
sfaces that meet G*. We call a* the weight associated with G*. In particular, the weight
associated with a facet F* of P* is a 1weight that is supported on the edges of P* that
meet F*, and so this must be the same weight described by McMullen in [
14
].
What is dual to the notion of the restriction of a n /  w e i g h t a* on P* to a facet F*? Let
a(x) be the dual (d  / )  s t r e s s on P and let vk be the vertex o f P corresponding to F*.
We could simply truncate a (x), eliminating the terms not supported on openstar Vk. This
would not necessarily be a stress on openstar Vk, but it would directly correspond to the
restriction of a* to F*. On the other hand, we can apply O/Oxk to a(x), which depends
only on the terms of a(x) that are supported on openstar Vk. This gives a (d  i 
1)stress on clstar vk. Projecting clstar vk onto a hyperplane orthogonal to vk, deleting Ok,and
applying Theorem 7 yields a simplicial (d  1)polytope F dual to F* and a (d  i 
1)stress dual to the restriction o f a* to F*.
We now have another way of interpreting McMullen's alternative formula [
13
], [
14
]
for multiplying by a 1weight a*. Remembering that we can view a linear istress on
P as either a polynomial o f degree i or an operator o f degree (d  i) as it suits us, we
choose to let a* correspond to an operator a = )~=1 rlj(O/Oxj) o f degree 1, and to let
any other given weight y* correspond to a polynomial y. Then
which can be regarded as a dual recasting o f McMullen's formula
n
ya = E
i=1
Oy
rlj Oxi'
n
where a*(F*) denotes the restriction of a* to the facet F*.
10.5. Shellings and Flips
McMullen [
13
] uses a shelling argument directly on the simple polytope P* to find a
basis for f2 and to prove that dim f2i = hi, i 0 . . . . . d. A general hyperplane is moved
"upward" through P*. When a vertex of type i is encountered (i.e., a vertex with exactly
i edges "below" the hyperplane), an arbitrary/weight can be assigned to t h e /  f a c e F
determined by these edges. The Minkowski relations defining the weights can be used
to find the unique/weights that must be assigned to t h e /  f a c e s that do not have a vertex
o f type i at the top.
It happens that this basis is not dual to the closed star stress basis constructed in
Section 8. For the dual basis, again a general hyperplane is moved upward through P*.
W h e n a vertex o f type i is encountered, we take F* to be the a c c o m p a n y i n g /  f a c e and add
into the basis for f2d_i the (d /)weight associated with F* as in the previous section.
In some sense the elements of this basis are more local than those o f McMullen's basis.
We conclude with some comments on McMullen's flips [
13
]. As he points out, flips
are dual to bistellar operations. Even though we are considering bistellar operations in
a more general context, it is straightforward to verify that our Theorem 13 is dual to
McMullen's Theorem 11.3 in [
13
], and that the justifications of these two theorems are
essentially the same in a combinatorial sense.
11.
In Section 5 we considered a matrix whose left nullspace defined affine kstress. The fight
nullspace then turned out to be an appropriate generalization of infinitesimal motions.
We can try the same procedure with kweights on a simple dpolytope P. For each
(k + 1)face G, consider a rigid motion ~oGthat maps affG onto (R k+l , 0 . . . . . 0) C R d
and then projects this space naturally onto R k+l. Construct a matrix R with one row
for each kface F of P and columns occurring in groups of k + 1, one group for each
(k + 1)face G. The row vector of length k + 1 in row F, group G, is
O r
goG(u~,a)
if
if
F f f . G,
F C G,
where UF,6 is the unit outer normal vector of F with respect to G in affG.
Define m = ~01 (m') (where ~o1 is interpreted in the obvious way) to be a (k +
1)circulation when m' is a member of the fight nullspace of R. So m is an assignment to
each (k + 1)face G of a vector parallel to G that satisfies the conditions
for every kface F. Denote the space of (k + 1)circulations by Ck+l.
In the case that k = 0, we have a vector, or flow, associated with each edge of P and
a condition on each vertex of P that forces flow conservation. For higher values of k we
can interpret the (k + 1)circulation as a translation of the (k + 1)dimensional content of
the (k + 1)dimensional faces in directions parallel to these faces with flow conservation
across every bounding kface.
Theorem 23. Let P be a simple dpolytope. Then dim C k + 1 ~ hk  f k I (k + 1)fk+l.
Proof. This is an immediate consequence of the fact that R is an fk x (k + 1)fk+l
matrix with a left nullspace of dimension hk. []
It is clear that dim C1 = h0  f0 + fl = fl  fo + 1, which is the dimension of the
space of ordinary circulations on a graph with f0 vertices and fl edges. In general, in
terms of the f  v e c t o r of the simple dpolytope P (the reverse of the f  v e c t o r of the dual
simplicial polytope)
So
h * =
Ed ( _ 1)jk(J)k 3~
j=k
= f k   ( k + l ) f k + l +
E
j=k+2
( _ l ) J  k
J
JJ"
dim Ck+1
d
E
j=k+2
(
k,k] j"
12. Unbounded Simple Polyhedra
Consider the boundary complex A of a simplicial convex dpolytope P with vertex set
vl . . . . . vn. Consider the ring B = Bo <9... <9 Ba = A/(O1 . . . . . 0n), where the Oi are
constructed from the vj as in Section 3.2. As we have already mentioned, the gtheorem
implies that hi = hdi, i = 0 . . . . . ld/2J and also that gi > gi1, i = 1. . . . . / d / 2 J ,
and this is proved by showing that multiplication by wd2i is a bijection between Bi and
Bd_i, i = 0 . . . . . /d/2J.
Now let v be any vertex of P and consider the simplicial complex E = A \ v . In [
3
]
it is proved that h i ( E ) > hd_i(E), i = 0 . . . . . [ d / 2 J , and also that h i ( E ) > h i + l ( E ) ,
i = Ld/2J . . . . . d. This is a consequence of the gtheorem, but now we can view this as
a consequence o f a weakened Lefschetztype theorem on the face ring o f E. Let At =
R[xl . . . . . xn]/l:: and B' = B~ < 9 . . . <9 B'd = A'/(O1 . . . . . Od). Take o: = Xl + . . . + xn
as before.
Theorem 24.
Multiplication by (0d2i is a surjectionfrom Bi to Bai, i = 0 . . . . . / d / 2 J .
It is more convenient to prove this with weights instead o f stress. Let P* be the simple
dpolytope dual to P and let F* be the facet o f P* corresponding to v. It can be arranged
(for example, by choosing the origin suitably close to v) that discarding the inequality
defining the facet F* results in an unbounded simple polyhedron Q* which is dual to
the simplicial complex E. We can define weights on Q* in the natural way, even though
Q* is unbounded. So dim [2i(Q* ) = h d  i ( E ) , i = 0 . . . . . d. What we actually prove is:
Theorem 25.
0 . . . . . l a / 2 j .
Multiplication by pd2i is an injection f r o m ~'~i(Q*) to ~'2d_i(Q*), i =
Proof. Use McMullen's construction to consider a basis o f f2(P*) determined by a
hyperplane. Choose this hyperplane so that it first moves past the vertices in F* before
it encounters the remaining vertices of P*; i.e., arrange for F* to be at the "bottom"
of P*.
Consider a vertex vj o f type i in P and the associated basis element a o f f2i (P*). In the
case that vj is also in F*, vj is also a vertex of type i in F* and the restriction o f a to F* is
an element o f the basis o f f2 (F*). Reversing this restriction gives an injection o f f2i (F*)
into ~'2i (P*). In the case that vj is not in F*, the restriction o f a to F* is zero. Notice that
the weights o f Q* correspond naturally to the weights o f P* that are zero on F*. Hence
we have an injection of f2i (Q*) into ~i(P*). Identifying f2i (F*) and f2i (Q*) with their
images in [2i(P*) yields the direct sum decomposition ~ i ( P * ) = ~2i(F*) <9~i(O*).
Looking at the description of the multiplication of weights in McMullen [
13
], it is
seen that multiplying p by a weight that is zero on F* results in a weight that is also zero
on F*. Therefore, since multiplication by pd2i is a bijection from f2i ( P ) = f2i (F*) <9
ff2i(Q*) to ~ d  i ( P ) = ~ d  i ( F*) <9 f2di(Q*), it must be an injection from ~2i(a* ) to
~'~di ( Q*). []
Acknowledgments
Many colleagues and agencies have provided me with both material and intellectual
stimulation and support for this work, which has been evolving over a period o f years.
In particular, the investigations which eventually led to the definition o f kstress were
motivated by Kalai's results on algebraic shifting and on stress. Most o f the results in
Sections 3, 4, 6  8 were developed in 19871988, but the basis for the formulas for the
coefficients rests in some calculations begun during the author's stay at the
Mathematical Institute of the Ruhr University in Bochum, 19841985, which was supported by a
fellowship from the Alexander von Humboldt Foundation. Jonathan Fine's suggestion
in Oberwolfach 1989 that the volumes o f the faces of the dual simple polytope should
play an important role led to the material in Section 9, mostly discovered during the
author's stay at DIMACS, Rutgers University in 19891990. The definition o f
infinitesimal motions (Section 5) resulted from conversations with Nell White during a visit to
the MittagLeffler Institute in early 1992. The author had the opportunity to probe the
connections with weights during a visit to Peter McMullen at University College London
in the summer o f 1994, supported by the Engineering and Physical Sciences Research
Council, and it was during this time that most o f the material in the last three sections
was developed. The author is grateful for stimulating conversations with a number o f
individuals including Marge Bayer, Louis Billera, Bob Connelly, Paul Filliman, Jonathan
Fine, Sue Foege, Robert Hebble, Gil Kalai, A. Khovanskii, Peter Kleinschmidt, Peter
McMullen, Rolf Schneider, Stewart Tung, Neil White, and Walter Whiteley.
1. D.W. Barnette , The minimum number of vertices of a simple polytope , Israel J. Math. , 10 ( 1971 ), 121  125 .
2. D. W. Bamette , A proof of the lower bound conjecture for convex polytopes , Pacific J. Math., 46 (I 973) , 349  354 .
3. L. J. Billera and C. W. Lee , The numbers of faces of polytope pairs and unbounded polyhedra , European J. Combin. , 2 ( 1981 ), 307  322 .
4. P. Filliman , Face numbers of plspheres, Manuscript , 1991 .
5. P. Filliman , Rigidity and the AlexandrovFenchel inequality , Monatsh. Math., 113 ( 1992 ), 1  22 .
6. W. Fulton , Introduction to Toric Varieties, Annals of Mathematics Studies , Vol. 131 , Princeton University Press, Princeton, NJ, 1993 .
7. G. Kalai, Characterization of fvectors of families of convex sets in R d . Part 1: Necessity of Eckhoff's conditions, lsrael J . Math., 48 ( 1984 ), 175  195 .
8. G. Kalai, Rigidity and the lower bound theorem. I, Invent . Math., 88 ( 1987 ), 125  151 .
9. B. Kind and P. Kleinschmidt , Schalbare CohenMacauleyKomplexe und ihre Parametrisierung, Math. Z., 167 ( 1979 ), 173  179 .
10. C.W. Lee , Some recent results on convex polytopes, Contemp . Math., 114 ( 1990 ), 3  19 .
11. C. w. Lee Generaized stress and mtins in Pytpes: Abstract Cnvex and Cmputatina T . Bistriczky P. McMullen , R. Schneider , and A. 1 . Weiss, eds., NATO AS1 Series C , Vol. 440 , Kluwer, Dordrecht, 1994 , pp. 249  271 .
12. P. McMullen , The maximum numbers of faces of a convex polytope , Mathematika , 17 ( 1970 ), 179  184 .
13. P. McMullen , On simple polytopes , Invent. Math. , 113 ( 1993 ), 419 444 .
14. P. McMullen , Weights on polytopes, Discrete Comput. Geom., this issue , pp. 363  388 .
15. P. McMullen and D. W. Walkup , A generalized lowerbound conjecture for simplicial polytopes , Mathematika , 18 ( 1971 ), 264  273 .
16. T. Oda, Convex Bodies and Algebraic Geometry , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) , Vol. 15 [Results in Mathematics and Related Areas (3)], SpringerVerlag, Berlin, 1988 .
17. U. Pachner, Shellings of simplicial balls and p.1. manifolds with boundary , Discrete Math. , 81 ( 1990 ), 37  47 .
18. G. Reisner, CohenMacaulay quotients of polynomial rings, Adv . in Math., 21 ( 1976 ), 30  49 .
19. R. P. Stanley , The upper bound conjecture and CohenMacaulay rings , Stud. Appl. Math., 54 ( 1975 ), 135  142 .
20. R. P. Stanley , Hilbert functions of graded algebras, Adv . in Math., 28 ( 1978 ), 57  83 .
21. R. P. Stanley , The number of faces of a simplicial convex polytope, Adv . in Math., 35 ( 1980 ), 236  238 .
22. T .S. Tay, N. White , and W. Whiteley , Skeletal rigidity of simplicial complexes, I, II , European J. Combin ., 16 ( 1995 ), 381  403 , 503  523 .
23. T .S. Tay, N. white, and W. Whiteley , A homological interpretation of skeletal rigidity , Manuscript , 1993 .