#### Diameter-extremal subsets of spheres

Discrete Comput Geom
G e~oDmtKiec~t,tmelpnnutca~tlo.ltued
M. Katz 0
0 Department of Mathematics, State University of New York , Stony Brook, NY 11794 , USA
We investigate those spherical point sets which, relative to the Hausdortt metric, give local minima of the diameter function, and obtain estimates which, in principle, justify computer-generated configurations. We test the new sets against two classical conjectures: Borsuk's and McMullen's.
Introduction
It is not entirely obvious that such sets actually exist, so here are two examples
o f pointwise extremal sets:
1. The set of vertices o f a regular (2k + 1)-gon inscribed in the circle S ~.
2. The set of vertices of a regular (n + 1)-simplex inscribed in S".
More examples can be obtained inductively on dimension by the pyramid
construction. Suppose y c S "-~ is pointwise extremal. We view Y as a subset of
R n+~ by imbedding S " - I c R " c R ~+~. Let diamE Y = m a x , ~ . , v l ] x - y [ ] be the
Euclidean diameter o f Y. Choose p c R "+~ so that, for every y 6 Y c R "~ ~, one
has l l p - y l I = d i a m E Y. Then the set Y u { p } , scaled by a suitable factor, is
inscribed in a unit sphere S" and is a pointwise extremal subset of S" (pyramid
over Y).
In the two-dimensional case we have the following.
Proposition 1. Pointwise extremal subsets of S ~ include two infinite families of
subsets with, respectively, 2 k + 2 and 3 k + 1 points for ever), k >-1. Each o f these
subsets has diameter less than 2¢r/3.
Here the diameter of y c S 2 is measured in terms of spherical distances, i.e.,
lengths of arcs of great circles.
Remark. I f y c S 2 is pointwise extremal then it is clear that Y lies in no open
hemisphere, hence diam Y ~-arccos(-~) with equality only for the set of vertices
o f a regular inscribed tetrahedron (see [
14
]).
Theorem 1. Suppose Y c S 2 is a pointwise extremal set with n points and with
diam Y = d < 27r/3. Then the number ofpairs ofpoints {x, y} in Ywith dist(x, y) = d
equals 2n - 2.
An antiprism over a square such that the diagonal of the square equals the
distance from a top vertex to the two farthest bottom vertices, has eight vertices
and only 12 maximal distances. Thus the diameter restriction is indispensable.
An analog of Theorem 1 also holds for sets which are approximately pointwise
extremal (a suitable definition is given in 4.6). This observation is the key to
justifying computer-generated examples.
The p r o o f of Theorem 1 will actually imply the following: the Euclidean
convex hull of y c R 3 is a polyhedron which is dual to itself (up to homothety).
Such objects were considered by Lovfisz [
11
].
The p r o o f of T h e o r e m 1 has a similar flavor to tt:.e proofs o f Borsuk's conjecture
for finite subsets o f R 3 (see [
2
], [
8
], [
9
], and [
16
]). Restricting the class o f subsets
to subsets of the sphere actually enlarges the family of known examples. In fact,
such a restriction enables one to apply a procedure o f successive approximation
to construct new sets (see Section 2). However, we found no counterexample to
the conjecture for R 4 a m o n g a few pointwise extremal subsets of S 3 (see also
Section 5).
Next we define the space D ( S " , c), where c > 0, of all finite subsets Y c S"
with diam Y < co The diameter function, defined on D( S", c), associates to every
Y ~ D ( S " , c) its diameter diam y c S". The Hausdorff distance distil defines a
metric on D ( S " , c).
Diameter-Extremat Subsets of Spheres
A set Y E D ( S n, c) is called a critical point o f the diameter function if
diam Y < d i a m V'+ o(distH( Y, Y'))
for all Y ' c S n. Equivalently, one may define a critical point Y by saying that
small perturbations of Y can only decrease diam Y quadratically in the size o f
the perturbation.
Similarly, Y ~ D ( S ~, c) is called a minimum of the diameter function (or,
briefly, a minimal set) if diam Y-< diam Y' for all sufficiently small perturbations.
Theorem 2. Ever), critical point of the diameter function on D ( S :,2rr/3) is a
minimal set on S 2.
Note that the n u m b e r 2rr/3 in Theorem 2 is the best possible, as the set of
vertices of an equilateral triangle inscribed in a great circle has diameter 27r/3
and defines a critical point of the diameter function which is not a minimal set.
Corollary 1. I f d <27r/3, there are only finitely many noncongruent minimal
subsets o r s 2 with diameter <~d.
We may view the diameter function as a "Morse function" on the space of
subsets. A Morse theory-type argument yields the following (see Section 6).
Corollary 2. There is exactly one (up to congruence) minimal set in each connected
component of D ( S 2, 27r/3).
Question 1 (due to U. Abresch). Given that a minimal set is always pointwise
extremal (Corollary 3.4), is the converse also true?
In Section 1 we describe a general construction of pointwise extremal sets and
prove Proposition 1. Section 2 introduces a notion of separability and explains
how to use the sets o f the previous section to generate new examples by computer.
In Section 3 we study the tiling of S 2 defined by a pointwise extremal set o f
diameter <2"rr/3, and prove Theorem 1. Section 4 extends T h e o r e m 1 to
approximately pointwise extremal sets.
Section 5 describes a way of rewriting an inequality due to Walkup [
17
]. This
inequality is a special case of McMullen's conjecture [12, p. 561], proved in [
15
]
and [
1
]. It relates the numbers of edges and vertices of a convex simplicial
4-polytope. We rewrite this inequality in the context of self-dual polytopes defined
by pointwise extremal subsets of S 3, and observe that the rewritten version involves
the n u m b e r of pairs of vertices at maximal distance.
In Section 6 we prove T h e o r e m 2 essentially by counting the degrees of freedom.
Section 7 provides sufficient conditions for the existence of a minimal set near
an approximately minimal set.
I came to consider minimal sets while thinking o f M. Gromov~s notion o f the
Filling R a d i u s (see [
6
] a n d [
10
]). To sum up briefly the c o n n e c t i o n between the
two, one could conjecture the following: the filling radius o f the complex
projective space CP k with sectional curvature between 4~and 1 equals h a l f the spherical
diameter o f the ( 2 k + 2 ) - p o i n t set o f Proposition 1.
The filling radius o f a R i e m a n n i a n manifold X is defined as the infimum of
all e > 0 such that X, isometrically i m b e d d e d in some other space, bounds in its
e - n e i g h b o r h o o d U~X; i.e., i ~ [ X ] = 0 where I X ] is the f u n d a m e n t a l h o m o l o g y
class o f X and i~ is i n d u c e d by the i m b e d d i n g X ~ U~X.
A construction in the Banach space L " ( C P '') shows that the filling radius is
n o greater than the n u m b e r given above. Missing from the p r o o f o f the opposite
inequality is geometrical control o f minimal subsets o f diameter <2~v/3 o f C P ' .
B. Katz o f the AI Lab o f M I T should have been a c o - a u t h o r o f this paper,
were it not for his self-effacing modesty. I thank him sincerely for dedication
and reliability prevalent in his p r o g r a m m i n g and c o m p u t i n g work to obtain new
exampl'es o f minimal sets. These examples are the sole justification for the material
o f Sections 2, 4, and 7. M y thanks are also due to the University Paris 6 for
hospitality during the a c a d e m i c years 1981-1983 when I obtained the results o f
Sections 1 and 3, and to R. Spatzier for some subtle suggestions.
I. The Digon Construction of Minimal Sets
We define a digon to be the convex region on the sphere b o u n d e d by two meridians
(great semicircles joining north and south poles). Let /3 < ~-/2 be the angle o f
the digon and let P be a point on one o f the two meridians, if/3 < r < {distance
f r o m P to the nearest pole}, then the circle o f radius r centered at P intersects
the other meridian in two distinct points. O u r preliminary construction is as
follows.
We start at the north pole a n d " w a l k " south with step r on alternating meridians.
We leave a mark P, at the ith step, starting with Po = north pole. In other words,
we p r o c e e d to partition the digon into isosceles triangles A, with side r, starting
with A~ which has two angles equal to/3 (Fig. 1). Note that the e v e n - n u m b e r e d
marks P0, P 2 , . . . all lie o n one meridian.
Let d, = dist(Po, P,). Then the increasing sequence {d,} satisfies the recurrence
relation cos r = cos d, cos d,+ ~+ sin d, sin d,+ ~cos/3 (theorem o f cosines [4, p. 25]).
If step r equals/3, it is clear that infinitely m a n y steps will be required to reach
the equator. As the construction depends c o n t i n u o u s l y on r, we m a y choose
values o f r :>/3 so that the last mark Pk north o f the e q u a t o r and the first mark
Pk +~ south o f it are equidistant from the equator, i.e., d~ + dk ~~= ~r. Then P=k+~=
south pole, and the resulting partition o f the digon is invariant u n d e r the central
s y m m e t r y a r o u n d its center.
1.1. P r o o f o f Proposition 1. The sets with 2k + 2 points are just pyramids over
( 2 k + l ) - g o n s (see the Introduction). They have rotational symmetry o f order
2 k + l . The sets with 3 k + l points have rotational symmetry o f order 3. To
construct them, we let fl = 7r/3. Consider the action of Z3 on S 2 by rotations by
multiples of 27r/3 around P 0 = n o r t h pole. Then the orbit of the set
{Po, P 2 , . . •, P2k} under the action of Z3 is a pointwise extremal subset of S 2 with
3 k + 1 points. Its diameter d = d2k satisfies d + r = 7r. Note that the orbit of Pk
(or Pk+Jif k is odd) is an equilateral triangle with side d. []
We will see in Section 6 that these sets are in fact minimal.
1.2. Remark. The limiting configuration of the (2k + 2) family is the set
consisting of the north pole and the circle at latitude - ~ r / 6 . The limiting configuration
o f the ( 3 k + 1) family is a countable set with three accumulation points on the
equator.
It would be interesting to study the limiting configurations of general pointwise
extremal sets of diameter <2~r/3.
Let/3 = arccos 1/n. Consider the n + 1 meridians passing through the vertices
o f a regular n-simplex inscribed in the equator S "-1 = S". On each meridian, take
the point at spherical distance d~ fror0 the north pole, i = 2, 4 , . . . , 2k. These
points, together with the north pole, define a pointwise extremal subset of S n
with nk + k + 1 points.
A similar construction was independently described by Lov~isz [
11
].
When n = 1 the above construction yields the ( 2 k + 1)-gons in S ~.
2. Separability and the Justification of Computer-Generated Sets
We show how to construct new pointwise extremal sets starting with those of
Section 1, by means of an iterative computer procedure. This procedure relies
on a property that may be called "separability."
2.1. Definition. Let y c S 2, and suppose diam Y < 2 7 r / 3 . We say that points
x, y ~ Y are separated by Y if there are points z, w E Y such that the set {x, y, z, w}
does not lie in any hemisphere. Y is called separable if every pair of points of
Y are separated by Y.
A pointwise extremal set is necessarily separable (see 4.1). Thus all sets of
Proposition 1 are separable. Moreover, the sets o f the (3k + 1) family have many
separable subsets.
If Y c D ( S 2, 2¢r/3), then the unique minimal set Y in the connected c o m p o n e n t
o f f" is pointwise extremal (cf. Corollary 2 of the Introduction and Corollary
3.4 of Section 3).
2.2. Lemma. Suppose ~"is separable. Then Y and ~'have the same number of points.
Proof Three points on S 2 define a convex spherical triangle. Four points on S 2
define four triangles whose sum is a 2-cycle. If the 4-tuple does not lie in any
hemisphere, this 2-cycte belongs to the generator o f H2(S ~-,Z2). The lemma follows
from the fact that if all pairwise distances are <2~r/3, then the 2-cycle depends
continuously on the four points and hence the homology class is fixed.
2.3. Question. We have defined a map ~: Y a - ~ z 2 which associates to each
4-tuple the corresponding element of H2(S 2, Z2). Does ¢ determine the minimal
set y c S 2 up to congruence?
To construct a new minimal set Y, we choose a separable subset Y of a
( 3 k + l ) - p o i n t set of Proposition 1. Let n = # Y. We replace D ( S 2, 27r/3) by a
finite-dimensional configuration space V defined as the n-fold Cartesian product
o f copies o f S 2. We may view Y as a point of V.
At every point of V, consider the " g r a d i e n t " vector pointing in the direction
o f greatest decrease o f the diameter function on V, and of length equal to the
partial derivative in that direction. We move Y along this vector field to a zero
o f the gradient field. Thus we obtain a critical point of the diameter function
which is a minimal set by Theorem 2. A recursive simulation of this procedure
can be done by computer. The recursion produces a set which is approximately
minimal, and Proposition 7.1 tells us when there is a genuine minimal set nearby.
I f we apply the recursive procedure which moves only one point of Y at a
time, then we may get a set which is approximately pointwise extremal but not
approximately minimal, in a sense explained in Section 7. In that case Proposition
Fig. 2
Point
7.1 is not applicable, and extra work would have to be done to show that there
is a pointwise extremal set nearby.
To give an example, we consider the set with 3k + 1 points of Proposition 1
for k = 3 (see 1.1). Remove the points P2 and R4~/3P6. The c o m p u t e r then
produces the set of Fig. 2, with the points labeled as in Table 1; the latter also
contains the exponential coordinates to three decimals. The graph in the figure
is defined by the tiling (cf. Section 3).
3. The Structure of Pointwise Extremai Sets
In this section we prove Theorem 1 and derive further properties of pointwise
extremal sets.
Suppose Y is as in Theorem 1. Let y c Y. Let Hy c S 2 denote the spherial
convex hull of the set {x c Y tdist(x, y) = d}. Since d < 2~r/3, H,, is a
nondegenerate spherical polygon. The polygon Hv is inscribed in the circle of radius ~ r - d
centered at y ' = the point o f S 2 opposite y. The vertices of Hy are precisely the
points o f Y at distance d from y.
3.1. Remarks. 1. For every vertex x ~ H,,, the vector xy' points inside Hr. This
is because, by assumption, y is held by Y.
2. No point o f Y can lie on a side of Hy.
Two adjacent vertices a, b of Hy determine a convex cone ay'b of directions
at y', leading to no other vertices of Hv. Such a cone is called a sector at y'. In
the antipodal picture, points a ' and b' define a cosector, i.e., a cone of directions
at y.
3.2. Lemma. Suppose Y c S 2 is pointwise extremal with diam Y = d < 27r/3. Then
the sphere is tiled by the polygons Hy:
U Hy : $ 2 ,
y~_Y
I?tynIYlx=O
if y C x .
Proof I f /~y n H~, is n o n e m p t y then some vertex z of H , lies in H~. (or vice
versa) and so d _ dist(y, z) > dist(y, Hy) = d.
S u p p o s e C is a c o n n e c t e d c o m p o n e n t o f the c o m p l e m e n t S~+\t._.J,. ~ H,. Then
C has c o n v e x b o u n d a r y OC. I n d e e d , let us show that any i n t e r i o r a n g l e c+ o f C,
with vertex x, is acute. Let H~, and H. be the p o l y g o n s a d j a c e n t to c+ (Fig. 3).
Then x is a vertex o f b o t h H,, and H by R e m a r k 3.l(2). Every t a n g e n t vector
in the cone w ' x z ' facing C is c o n t a i n e d in either H~,, C, or H=. Hence by R e m a r k
3.1(1), w ' x z ' is a cosector. Thus o+ lies in a c o s e c t o r a n d is acute.
It follows that OC is simple. To see this, s u p p o s e we a p p r o a c h the b o u n d a r y
o f C in two disjoint n e i g h b o r h o o d s in its interior. C o n s i d e r the shortest curve
in C j o i n i n g the two b o u n d a r y points. Because OC is convex, this curve lies in
the interior of C a n d t h e r e f o r e is a m i n i m i z i n g arc o f great circle. H e n c e the two
b o u n d a r y p o i n t s are distinct as points on S-'.
Similarly, one can show that C is s i m p l y c o n n e c t e d a n d hence OC is c o n n e c t e d .
We view OC as a p o l y g o n a l curve a n d label its vertices a~ . . . . . a,,, (m.>-3) so
that the side a , a , ~ c H , = H , , (Fig. 4) (the indices are m o d m). C o n s i d e r the
s i m p l e c l o s e d p o l y g o n a l curve E = aly'~a2y~ • " • a,,,y',,,. The angles o f E facing <5"
are a l t e r n a t i n g sectors a n d cosectors. H e n c e E is a convex curve. It is easy to
see that a convex simple closed curve on S 2 has length at most 2~r (with equality
only for a digon described in Section 1). On the other hand, length E = 2mr >-6r >
27r, where r = ~ - d. The contradiction shows that the H,. tile is S 2. []
3.3. Corollary. Suppose y c S 2 is pointwise extremal with diam Y = d < 27r/3.
Then:
1. For every point p c S 2, there is a point y ~ Y with dist(p, y) >-d, with equality,
(for the farthest y) if and only if p c Y.
2. No proper subset Y ' c y with diam Y' = d is pointwise extremal.
The p r o o f is obvious. In particular, the graph whose vertices are points of Y and whose edges correspond to pairs o f points of Y at distance d, is connected. Corollary 6.3 will provide a more detailed description of this graph.
3.4. Corollary. Suppose Y ~ D( S 2, 27r/3) is a critical point o f the diameter
function. Then Y is pointwise extremal.
Proof Note that without the diameter restriction Y may not be pointwise
extremal, e.g., the 4-point set Y4 consisting of the north pole and the vertices of
an equilateral triangle inscribed in the equator (of course, diam Y4=2rr/3 so
Y4~ D ( S 2, 27r/3)).
Let d = d i a m Y. Let YoC Y be defined by Y o = { y 6 Y[diamy Y = d } , where
diamx Y = - m a x ~ v dist(x, z). The set Yo6 D ( S ~, 27r/3) is a critical point of the
diameter function but may not be pointwise extremal. Note that if y ~ Yo is not
held by Yo (see the Introduction) then we can perturb y so as to decrease diamy Yo
linearly in the size o f the perturbation.
We define the sets Y~ recursively by
Y~= {y~ Y~_~ly is held by Y~_~},
i = t , 2 , . . . .
The sequence must stabilize before we exhaust Yo, for otherwise we could
construct a perturbation linearly decreasing diam Yo, contradicting the
hypothesis. The stable set y ' c y is both pointwise extremal and a critical point
o f the diameter function, and hence Y ' = Y by Corollary 3.3(1). []
Proof o f Theorem 1. The spherical graph defined by the tiling has n vertices and
n faces. Hence the graph has 2n - 2 edges by Euler's theorem [4, p. 15]. A vertex
v with valence k lies opposite a k-gonal face f Let e be the number of edges
and m the number of pairs of points at distance d. Then 2 e = ~ k ( v ) while
2m = 2 k ( f ) . Therefore m = e = 2 n - 2 . The theorem is proved. []
Approximately Extremal Sets
We prove an analog of Theorem 1 for sets which are almost pointwise extremal.
This analog (Proposition 4.6) is proved under the hypothesis that sufficiently
m a n y of the pairwise distances are approximately equal, and that in addition the
set is separable in the sense o f 2.1.
4.1. Lemma. Suppose y c S 2 ispointwiseextremalwith diam Y = d <27r/3. Then
Y is separable.
Proof Let x, y e Y. Let a'xb', where a, b ~ Y, be the cosector (see 3.1) containing
the vector xy. If the set {x, y, a, b} is contained in some hemisphere, then x is
contained in the (convex) triangle bay (Fig. 5). Therefore by + ay > bx + ax = 2d,
and hence either by > d or ay > d. The contradiction proves the lemma.
4.2. Corollary.
the points o f Y by
Let Y e D ( S 2, 27r/3). Let x, y ~ Y. Define a relation ~ among
x ~ y
¢¢> ~ ( x , y , z , w ) = O
f o r a l l
z, w c Y
(cf. 2.3). Then ~ is an equivalence relation.
Proof. Clearly, x ~ y if and only if x and y collapse to the same point of any
pointwise extremal set in the connected c o m p o n e n t of Y c D ( S 2, 2~r/3) (cf. 2.1).
4.3. L e m m a . Suppose points Pi ~ S 2, i = 1, 2, 3, 4, do not lie in any hemisphere.
Let ~ / 2 -< d < 27r/3, and a s s u m e dist(P,, Pj) -< d f o r all i, j. Then
COS2 d
cos dist(Pl, P:)-< 2 cos2(d/2 ) 1,
with equality if and only if triangles P~P3P4 and P:P3P4 are equilateral with side d.
To prove Lemrna 4.3 we will need the l e m m a below which will also be useful
in Section 7 (see 6.6).
4.4. Definition. Denote by c~(d) the angle of the equilateral spherical triangle
with side d. Then 2 c o s ( d / 2 ) sin ( a ( d ) / 2 ) = 1.
Proof.
By the theorem of cosines
c o s / _ B A C =
cos B C - c o s A B cos A C
sin A B sin A C
>
cos d
sin A B sin A C
cot A B cot AC.
Since cos d <--0, we have
Proof o f L e m m a 4.3. Denote by U the point of intersection of the shortest arc
P3P4 and the great circle through P~ and /'2, so that UP~+P, P2+P2U=2~r.
To prove the lemma it suffices to show that UP~ is less than the height o f
the equilateral spherical triangle with side d. By first variation, UP1
<max(P4Pj, P3P~, h), where h is the length o f the perpendicular dropped from P~
to (possibly the continuation of) t'31"4. We may assume that h >--P3P~; then the
angle/-t"1t"3t"4 is obtuse. Spherical trigonometry gives
sin h = s i n P~P3 sin (/_P~P3Pn)>-sin d sin a ( d )
by L e m m a 4.5. The l e m m a is proved.
Together, Lemmas 4.1 and 4.3 show that the distance between two points of
a pointwise extremal set can be bounded from below in terms of the diameter
o f the set. Thus the n u m b e r of points n in such a set can be b o u n d e d from above
in terms o f d by a volume argument. It is not clear if the converse is true: can
d be b o u n d e d away from 27r/3 in terms o f n?
Let C be a collection of pairs of points of a set y c S 2. In analogy with the
definition given in the Introduction, we say that a point y e Y is held by C if,
for every tangent vector v e TyS 2, there is a pair {x, y} e C such that v. yx-> 0. By
abuse o f notation, we will not distinguish between a pair of points and their
distance.
4.6. Proposition. Let Y ~ S 2, # Y = n , diam Y = d = 2 ~ r / 3 - t , where t > 0 . Let
d~ >-. • • >-d , be the m largest distances among points o f Y. Suppose the following
three conditions are satisfied:
1. Y is separable (cf. 2.1).
2. Each point o f Y is held by the collection {dk}, k = 1 , . . . , m.
3. d ~ - d , < t .
Then m = 2 n - 2 .
4.7. Corollary. I f d ~ - d m < t / 2 then d , - d m + ~ > t/2, i,e., the ( 2 n - 1 ) t h largest
distance d2,-1 is " m u c h " smaller than d2,-2.
P r o o f o f Proposition 4.6. Let y ~ Y. To define an analog of H~ we consider the
vectors yx, where {x,, y} ~ {dk}, ordered clockwise, and construct a polygon H,
by joining the consecutive points x, by shortest arcs. Note that /2/~.c ~ / 4 : - - G if
y ~ z, for otherwise a vertex w of /4_. must lie in /2/~ (or vice versa) and hence
dist(y, w) > dist(y, H~,) >- d,,, so that w must must be a vertex of H~. We will show
that the polygon H,,, while no longer inscribed in a circle, is convex. Then the
proposition follows from an analog of Lemma 3.2 by the argument of the proof
o f Theorem 1.
To see that /4,, is convex, it suffices to show that each of the angles Z_y'x,x, ~
and/_y'x,x,__~ is acute. Let x, = a and x,+~ (or x,.~) = b. We have
c o s / - y ' a b =
cos y ' b - cos y ' a cos ab
sin y ' a sin ab
To show that this is positive, we estimate the numerator using Lemma 4.3:
cos y' b - cos y ' a cos ab = cos ya cos ab - cos y b
cos d~ cos ab - cos d,,
= cos dl(cos a b - 1 ) - (cos d m - cos d~)
cos dl (cos ab - 1) - ( d~ - d,, )
->cosd
(
c ° s 2 d )
2 c o s 2 ( d / 2 ) 2 - t .
Since arccos(-~) -< d < 2~r/3 and arccos(1/~3) -< d / 2 < rr/3 (see the Remark
following Proposition 1 o f the Introduction), we have
~os~ ~ ~ o~ so~~, _~)_2~_co~, c o s1~ , (co~ c o ~ ) ( c o ~~+ ~o~~)
~_~ ~, ~3, ~, (cos ~~ -1- COS ~)
= cos \(-~~ - ~ /t~+cos(3
,)
1 t v / 3 . t l
=-2cos~+T sm ~ - ~ cos ,+Tsin,
- > ~ ( s i n ~ + s i n t )
-~~,f0 (~co~+~os,)~,
> _ _ ~ f ' 3
- 2 J o 2 c ° s t d t
3x/~ x/3
" - - t
4 2
< t ,
as t < 2"n'--~= ~']
3 2 6 /
Thus H~, is c o n v e x a n d the p r o p o s i t i o n is p r o v e d .
Waikup's Inequality and Minimal Subsets of S 3
F1
Here c is the n u m b e r o f cells o f P, f is the n u m b e r o f faces o f P, a n d a I is the
n u m b e r o f j - g o n a l faces o f P.
N o w we a p p l y the p r o c e d u r e of p u l l i n g the vertices o f P (see p. 116 of [
13
])
to c o n s t r u c t a s i m p l i c i a l p o l y t o p e Q with the s a m e n u m b e r o f vertices v as P.
Let E and e be the numbers of edges o f Q and P, respectively. The description
of Q given in [
13
] allows one to write down the following formula for E:
E = e + (a 4+ 2a5 +" • ") + "cell diagonals,"
where cell diagonals (c.d.) are the edges joining pairs of vertices of Q which did
not belong to any c o m m o n face of P (not all such pairs need be joined by an
edge of Q). Therefore ~,. vc = 2c + f + E - e - c.d.
Walkup [
17
] proved that E - 4 v - 10. (This is a special case o f McMullen's
conjecture proved by Stanley [
15
] and Billera and Lee [
1
].) Hence ~, v
c>2 c + f - e + 4 v - l O - c . d .
Finally, if P is as in 5.1, then f = e and v = c = n. Also, ~c v~ = 2N, where N
is the number of pairs of points at maximal distance. Hence
N > _ 3 n - 5 - 1 c . d .
For example, the pyramid over the set with 3k + 1 points for k = 3 of Proposition
1 (see.Section 1) has n = 11 and N = 3n - 5 = 28. But it also has the property that
any nearby simplicial polytope must possess cell diagonals. Hence it is not
extremal for Waikup's inequality.
Caratheodory and the Proof of Theorem 2
Let D~(S 2, d ) c D ( S : , d) denote the space of separable subsets of S 2 (see 2.1).
It follows from Corollary 4.2 that Y e D is a critical point of the diameter function
on D if and only if it is a critical point of its restriction to De.
On the other hand, D~ locally imbeds in the n-fold Cartesian product V =
S 2x . • • x S 2, with typical point Y = (y~ . . . . . y,). Denote by 7r,: V ~ S 2the
projection to the ith factor. Consider the function f j on V defined by f ~ ( Y ) =
dist(Tri(Y), 7r~(Y)). Clearly,
Twrr,(Vfo)=y,yj and
Ty~r,(Vfmj)=O
if m # i .
(*)
Here y z c TyS 2 denotes the unit tangent vector to the shortest arc joining y
with z.
If y c S 2 is pointwise extremal with diam Y = d, then there are 2n - 2 pairs
( i , j ) such that f j ( Y ) = d by Theorem 1. We relabel these functions by fk,
k = 1 , . . . , 2 n - 2 .
The orthogonal group SO(3) acts on V by g . Y = (gYl . . . . , gy,,). The orbit of
Y under this action is three-dimensional as long as Y contains a pair of distinct
nonantipodal points. Let N = N y c T y V be the tangent space to the orbit.
T h e d f k c T * V satisfy dfk(N)=-O, and hence span at most a ( 2 n
3)-dimensional subspace of T* V. Thus they are linearly dependent:
Let a(y,, )'/) be the coefficient off~ in this sum. By (*), equation (**) is equivalent
to the following system o f equations:
h, tt~
a(h, y)yh = 0
for every
y c Y,
(***)
where the s u m m a t i o n is taken over all vertices o f H~ (see Section 3).
6.1. Lemma. Let Y be a pointwise extremal set. Suppose the coefficients in (**)
sati.~V a~ > 0 f o r all k. Then a~ > 0 Jor all k.
We will need the following extension o f the results o f Section 3 to prove
L e m m a 6.1.
6.2. Lemma. Let y c S 2 be a pointwise extremal set with diam Y = d < 2 7 r / 3 .
Let x, y ~ Y with dist(x, y ) = d. Then either x is not held by Y\{y} or y is not held
by r\{x}.
Proof Let a and b be the vertices of the face H, which are adjacent to x. If v
is held by Y \ { x } then the union of the sectors a y ' x and xy'b is still a convex
cone at y'. By k e m m a 3.2 there is a face H= bordering H~ on the side ax, and a
face H~. bordering H,. on xb (Fig. 6). If x is held by Y\{y} then the u n i o n of
the cosectors z ' x y ' and y ' x w ' is a convex cone. Thus the sum of the four angles
involved is <2rr.
On the other hand, the angles o f the " d i a m o n d " x z ' a y ' satisfy L z ' x y ' + L x v ' a >
7r, and, similarly, L w ' x y ' + / _ x y ' b > 7r. The contradiction shows that each pair o f
points at distance d is indispensable for the pointwise extremality o f Y. We are
grateful to B. Heepe for pointing out that the diameter restriction should not
play an essential role in the proofl 5
Fig, 6
6.3. Corollary. Let {xi, Yi}, i = 1, 2 with dist(x~, y~)= d, be unordered pairs o f
points o f Y. Then there is a sequence o f ordered pairs Pk, k = 1, . . . , m, such that
the following f o u r properties are satisfied:
1. p~ is an ordering o f {x~, Yl} and Pm is an ordering o f {x2, Y2}.
2. end(pk) = init(pk +~) , k = 1. . . . . m - 1, where end(x, y) = y and init(x, y) = x.
3. e n d ( p k ) is not held by Y \ { i n i t ( p k ) } , k = 1 , . . . , m - 1.
4. m <- n = the number o f points o f Y.
This follows from L e m m a 6.2 and C o r o l l a r y 3.3(2).
[]
Proof o f L e m m a 6.1. Since (**) is equivalent to (***), it suffices to show that
if a(x, y ) = 0 for some x, y e Y, then all the coefficients must vanish.
I f y is not held by Y \ { x } , then the vectors yh, h e H~.\{x} all lie in an o p e n
halfspace. Since all ak >--O, (***) implies a(h, y ) = 0 for all h c H~.. N o w the l e m m a
follows from C o r o l l a r y 6.3. CI
Proof o f Theorem 2. I f Y e D ( S 2, 2 ~ / 3 ) is a critical point o f the diameter function
then by a first variation a r g u m e n t there are coefficients ak -> 0 such that ~ ak d fk = 0
at the point Y c V. (Recail that V = S2x . . . x S 2, a n d N c T v V is the tangent
space to the SO(3)-orbit.) Equivalently, for every v ~ T v V there is an fk such
that dfk(v)>--O. To show that Y is a minimal set it suffices to prove that, for every
v e T v V \ N , there is a function fk with d f k ( v ) > O . Suppose dfk(v)<--O for all k.
By projecting to the line s p a n n e d by v we see that, for every k, either dfk(v) = 0
or ak = 0 . By C a r a t h e o d o r y ' s theorem (see p. 26 o f [
13
]) there are coefficients
v2n -2
bk>--O such that ~k=~ b k d f g = O , where one o f the coefficients bk equals 0. But
this contradicts L e m m a 6.1. The theorem is proved. []
Proof o f Corollaries 1 and 2 o f the Introduction. Corollary 1 is immediate from
the c o m p a c t n e s s o f V.
S u p p o s e two minimal sets lie in the same c o n n e c t e d c o m p o n e n t o f
D~(S 2, 2~-/3). Then they also lie in the same c o n n e c t e d c o m p o n e n t A o f D~(S z, d)
for some d < 2 ~ / 3 . By C o r o l l a r y 1, we can c h o o s e a minimal set Y c A with the
biggest diameter. Let Z ~ A be another minimal set, Let
a = inf{c < d ] Y a n d Z lie in the same c o n n e c t e d c o m p o n e n t o f D~(S 2, e)}.
We have a > diam Y because Y is a minimal set. Let e = ½m i n ( d - a, a - diam Y).
Let 7 be a curve in D s ( S 2, a + e) joining Y and Z a n d transverse to the S O ( 3 )
action.
There are n o critical values in the interval [ a - e, a + e]. We construct a s m o o t h
n o n v a n i s h i n g vector field which forms an acute angle with the " g r a d i e n t " o f the
diameter function (cf. Section 2). This is d o n e by p a t c h i n g together radially
parallel fields s u p p o r t e d in small balls as in the p r o o f o f the I s o t o p y l e m m a o f
[
7
] a n d [5, p. 182] for the distance function. We can push the curve 3' into
D~(S 2, a - e ) by following this field, contrary to the definition o f a. This p r o o f
o f C o r o l l a r y 2 is spiritually akin to the m o u n t a i n pass l e m m a o f Morse t h e o r y
(see L e m m a 4.11(2) o f [
3
]). []
Remark. All the pointwise extremal sets o f Proposition 1 are minimal subsets
of S 2. To see this we must show that all the coefficients in (**) have the same
sign. In the case of the sets of Proposition 1, this follows easily from the following
two observations:
1. If H~. is a triangle then all three coefficients a(h, y), h ~ Hv, have the same
sign.
2. S u p p o s e Hy is a quadrilateral with vertices h,, i = 1 , . . . , 4, labeled so that
y is held by {h~, h2, h3} and by {h~, h2, h4}, but not by the remaining two
triples. S u p p o s e that sign a(h3, y) = sign a(h4, y). Then all four signs are
the same.
In Section 7 we will need the following two estimates. Let a ( d ) be defined as in
4.4.
6.4. Lemma. L e t y c S 2 be a minimal set, and let n = # Y, d=diam Y < 2 ~ - / 3 .
Then the coefficients of (**) satisfy
min~ ak _>( c o s a-~d)) "-~.
maxh Ok
Proof Let y ~ Y. A s s u m e y is not held by Y \ { x } , and let z and w be the vertices
o f H,. a d j a c e n t to x. Let v ~ T,S 2be the bisector of/_zyw. Then for every h ~ H~.\{x},
the vectors v and yh define an angle - < a ( d ) / 2 at y, because dist(z, w ) -< d. Let rr
be the o r t h o g o n a l projection of TvS 2 to the line s p a n n e d by v. Then ttTr(yh)ll
>c o s ( a ( d ) / 2 ) . S u p p o s e for s o m e hoe H~. we have a(ho, y) cos ( a ( d ) / 2 ) > a(x, y),
where the coefficients are as in (***). Then
-->I[a(ho, y) 7r(yho) + a(x, y) 7r(yx)l[
The contraction shows that a(h, y) cos ( a ( d ) / 2 ) <_ a(x, y) for all h E H~.. N o w the
l e m m a follows from C o r o l l a r y 6.3. D
T h e s a m e reasoning p r o v e s the following.
6.5. Lemma. Let y c Y and assume a coefficient bh ~ 0 is assigned to each h ~ H,,.
Suppose y is not held by Y \ {x}. I f bx < ½(cos( a ( d )/ 2 ) ) b~, , for some ho c Hy, then
I1~,bhyhll > ~(cos(a( d)/2))bho.
6.6. S u p p o s e Y is a p p r o x i m a t e l y pointwise extremal, in the sense that it satisfies
the hypotheses o f Proposition 4,6. The results o f Section 3 are still valid (with
the exception o f L e m m a 3,3(1) and 3.4). So are the results o f this section up to
L e m m a 6.4. Finally, L e m m a s 6.4 and 6.5 remain true in view o f the angle
c o m p a r i s o n L e m m a 4.5.
Approximately Minimal Sets and the Implicit Function Theorem
In Section 2 we described a c o m p u t e r p r o c e d u r e for obtaining minimal sets. The
object o f this section is to find estimates to justify that procedure. We will find
conditions on an a p p r o x i m a t e l y minimal set which guarantee that there is a
genuine minimal set nearby.
Since it is u n k n o w n w h e t h e r every pointwise extremal set is minimal, we have
to assume that the coefficients ak in equation (**) o f Section 6 have the same
sign. tt follows by 6.6 and L e m m a 6.4 that the coefficients are b o u n d e d away
from 0. Thus such a condition on the ak is computer-verifiable.
We also have to require a m o r e stringent inequality than 4.6(3) on the maximal
distances. This is because we use an implicit f u n c t i o n - t y p e argument, which is
o f a different kind than the results o f the previous sections.
S u p p o s e Y c S 2 has n points and diam Y < 27r/3. We order the set o f all pairs
o f points o f Y by decreasing distance: diam Y = d,-> d 2 >- . • •-> d2,, _2>>-d2,,-,,
etc. By abuse o f notation we will not distinguish between a pair o f points and
the distance between them. C h o o s e 2rr/3 > d o > d~. Let c~(d) be defined as in 4.4.
7.1. Proposition.
three conditions:
Suppose a set y c S 2 with diam Y < 27r/3 satisfies the following
1. Y is separable.
2. There are numbers bk ~ 0, k = 1. . . . . 2n - 2, such that, f o r every y e Y, we
have ~ b( h, y)yh = 0 , where the s u m m a t i o n is taken over all h e Y such that
.~f h , y } ~ { d k } k2=n ,-2 and b ( h , y ) = bk i f { h , y } = dk.
3. d, - d2,-2 < A, where
A - x / ~ n _ 2 ( n _ l ) 2 , _
~
c,
and c = ~ m i n ( d o - d~, d2,-2 - d2,-1).
Then there is a m i n i m a l set Ym Hausdorff-near Y, i.e., distil( Y, ~ ) <
pairs o f points at m a x i m a l distance correspond to the pairs d~ , . . . , d2,-2.
c, whose
N o t e that the difference d 2 , - 2 - d 2 , - , in the definition o f c is not essential as
it can be b o u n d e d from below in terms of, say, d o - d , (cf. 4.7). N o t e also that
the affirmative answer to Question 1 o f the I n t r o d u c t i o n w o u l d p r o b a b l y allow
one to replace 7.1(3) with 4.6(3).
Proof Let f : V ~ g 2" 2 be given by coordinate functions f~ . . . . ,f2,-~ defined
in Section 6. Then f ( Y ) lies " n e a r " the diagonal line L c R 2" -2. Let 7r: R 2 " - 2 ~ L ~
be the orthogonal projection to the complement of L. We will show that ~ - o f
maps a ball of radius c at Y ~ V (where V is endowed with the Riemannian
metric o f the Cartesian product) onto a neighborhood containing a ball of radius
. , / ~ - 2 A at ~r of(y). This will show that all t h e f k take the same value at some
Y,, E V because the distance from f ( Y ) to L is < , f ~ - 2 ( d , - d2,-2).
Condition 2 implies that each point of Y is held by {dk} in the sense of the
paragraph preceding Proposition 4.6. By 6.6 and L e m m a 6.1, bk > 0. Since the
ball B(Y, c) at Y c V of radius c is connected, it follows that Ym is a minimal
set. This proves the proposition because distil--< distv. []
We define the contracting coefficient of a linear m a p g: R p ~ R q to be
min
tlg(v)ll
Let Z c B( Y, c). We will estimate separately the contracting coefficients of T z f
( L e m m a 7.2) and 7r ( L e m m a 7.3) restricted to suitable subspaces. The proposition
follows by multiplying these coefficients.
7 . 2 . L e m m a . Let ~r[,mrt be the restriction of 7r to the image of T f Then the
contracting coefficient of Tr[lmTrr is >-(cos(a(do)/2)) "-1.
Proof Consider a slice for the action of SO(3) on V. The m a p f imbeds the
slice in R 2n-2. The image of Tf is parallel to the hyperplane a ~ x ~ + . . . +
a2~-~x2,_2 = 0, where the ak are as in (**). The contracting coefficient equals the
cosine of the angle between the normal vectors o f I m T f and L l :
a l + ' ' ' + a 2 n - 2
2nx/~ff~-2x/a~+ • . i+ a2n2_ 2
~_ m i n k a k ( ~ c o s ~ ) ' - t
maxkak
The last inequality is true by 6.6 and L e m m a 6.4.
D
7.3. Lemma.
The contracting coefficient S of TfIN~ satisfies
( n - 1 ) 2 "-1 c o s ~
.
Let v ~ N ± c TzV, IIv[I= 1. Let tr: N J - ~ v ~ b e the p r o j e c t i o n to the h y p e r p l a n e
v Z c N ~ o r t h o g o n a l to v. By C a r a t h e o d o r y ' s t h e o r e m we m a y c h o o s e an i n d e x
k~ a n d n u m b e r s bk >--0 so t h a t bk, = 0 a n d ~ bh~r(dfk)= 0. H e r e T V a n d T* V a r e
i d e n t i f i e d u s i n g t h e R i e m a n n i a n m e t r i c . T h e r e f o r e Y~bk dJ~ is p r o p o r t i o n a l to v
a n d IIZ bk dfktl = }E bk df~(v)t. Let bk~ b e t h e b i g g e s t o f t h e coefficients. C h o o s e a
s e q u e n c e o f m c o v e c t o r s j o i n i n g dfk, with dfk2 as in C o r o l l a r y 6.3. R e l a b e l the
c o v e c t o r s so t h a t k~ = t a n d k2 = ra, a n d so t h e s e q u e n c e b e c o m e s df~ . . . . . dfm.
H e r e b~ = 0 a n d m -< n.
I f 0 < e < 1, let i be the b i g g e s t i n d e x s u c h t h a t eb, > b,_~ (i ~ 2 s i n c e b~ = 0).
T h e n bk_~/bk >-e f o r all k > i, a n d
b~
= bi x
bi+l
. . . x
b . , - t b . , _
bm > e
" - . . . . . . .
bm>-e
2
b,,.
Let e = ½c o s ( a ( d o ) / 2 ) . T h e n b y 6.6 a n d L e m m a 6.5 we h a v e [l~ bk dfk [I-> eb, >_
em-~bm. S i n c e [[Y~bk dfk [[ = 12 bk dA(v)l, t h e l a r g e s t s u m m a n d [b~ dfe(v)l satisfies
Ibedfe(v)[>--(1/(m-1))~-'b,. ( s i n c e b ~ = 0 ) . But b.,>-be, t h e r e f o r e [dfe(v)[>_
( 1 / ( m - 1))e " - l , a n d so a l s o
- - >
m - 1
e
>-( n - l ) 2
"-~
c o s
4.5. Lemma. Let A B C be a spherical triangle with sides o f length <-d, and assume that sides A B and A C are >-7r/2 . Then Z. BAC < -or(d).
1. L. J. Billera and C. W. Lee , A proof of the sufficiency of McMullen's conditions for.fivectors of simplicial convex polytopes , J. Combin. Theory Ser. A 31 ( 1981 ), 237 - 255 .
2. K. Borsuk , Drei S/itze fiber die n-dimensionale euklidische Sphare, Fund . Math. 20 ( 1933 ), 177 - 190 .
3. J. Cheeger and D. G. Ebin , Comparison Theorems in Riemannian Geometry, North-Holland, Amsterdam, 1975 .
4. L. Fejes T6th , Lagerungen in der Ebene auf der Kugel und im Raum, Zweite Auflage , SpringerVerlag, Berlin, 1972 .
5. M. Gromov , Curvature, diameter and Betti numbers, Comment. Math. Hetv. 56 ( 1981 ), 179 - 195 .
6. M. Gromov , Filling Riemannian manifolds, J. Differential Geom. , 18 ( 1983 ), 1 - 147 .
7. K. Grove and K. Sbiohama , A generalized sphere theorem , Ann. of Math. 106 ( 1977 ), 201 - 211 .
8. B. Griinbaum , A proof of V~izsonyi's conjecture , Bull. Res. Council Israel A 6 ( 1956 ), 77 - 78 .
9. A. Heppes , Beweis einer Vermutung yon A. V~izsonyi , Acta Math. Acad. Sei. Hungar . 7 ( 1956 ), 463 - 466 .
10. M. Katz , The filling radius of two-point homogeneous spaces , J. Differential Geom . 18 ( 1983 ), 505 - 511 .
11. L. Lov~isz,Self-dual polytopes and the chromatic number of distance graphs on the sphere , Acta Sci. Math . 45 ( 1983 ), 317 - 323 .
12. P. McMullen , The numbers of faces of simplicial polytopes , Israel Z Math. 9 ( 1971 ), 559 - 570 .
13. P. McMullen and G. C. Shephard , Convex Polytopes and the Upper Bound Conjecture, London Mathematical Society Lecture Note Series , Vol. 3 , Cambridge University Press, London, I971 .
14. J. Molmir, [ Jber eine 1Jbertragung des Hellyschen Satzes in sph~irische R~iume , Acta Math. Acad. Sci. Hungar . 8 ( 1957 ), 315 - 318 .
15. R. Stanley , The n u m b e r o f faces of a simplicial convex polytope, Adv . in Math. 35 ( 1980 ), 236 - 238 .
16. S. Straszewicz, Sur un probl~me g6om6trique de P. Erd6s, Bull Acad. Polon. Sci. CI. III 5 ( 1957 ), 39 - 40 .
17. D. W. Walkup , The lower b o u n d conjecture for 3- and 4-manifolds , Acta Math. 125 ( 1970 ), 75 - 107 .