On Sumsets and Convex Hull
Discrete Comput Geom
Károly J. Böröczky 0 1 2 3
Francisco Santos 0 1 2 3
Oriol Serra 0 1 2 3
0 F. Santos Departamento de Matemáticas, Estadística y Computación Universidad de Cantabria , Av. de los Castros 48, 39005 Santander , Spain
1 K. J. Böröczky Central European University , Nador u. 9, Budapest 1051 , Hungary
2 K. J. Böröczky Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences , Reáltanoda u. 1315, Budapest 1053 , Hungary
3 O. Serra Departamento de Matemàtica Aplicada 4, Universitat Politècnica de Catalunya , Jordi Girona 1, 08034 Barcelona , Spain
One classical result of Freiman gives the optimal lower bound for the cardinality of A + A if A is a ddimensional finite set in Rd . Matolcsi and Ruzsa have recently generalized this lower bound to  A + k B if B is ddimensional, and A is contained in the convex hull of B. We characterize the equality case of the MatolcsiRuzsa bound. The argument is based partially on understanding triangulations of polytopes.
Sumsets; Shellable triangulations; hvector

1 Introduction
The topic of this paper is the cardinality of the sum of finite sets in the real affine
space. For thorough surveys and background, consult Ruzsa [6], and Tao and Vu [9].
A set A in Rd is ddimensional if it is not contained in any affine hyperplane. One
seminal result proved by Freiman [1] is that for any finite ddimensional set A in Rd ,
 A + A ≥ (d + 1) A −
.
(
1
)
The inequality is tight and the extremal sets have been characterized by Stanchescu
[7]. It was recently generalized by Matolcsi and Ruzsa [5] as follows.
Theorem 1 (Matolcsi–Ruzsa) If B is finite ddimensional in Rd and A ⊂ [B], then
for every k ∈ N
d + k
k
.
In particular, taking A = B they get the following, of which (
1
) is the case k = 2:
Corollary 2 (Freiman–Matolcsi–Ruzsa) If A is finite ddimensional in Rd , then for
every k ∈ N
We call such a pair ( A, B) a kcritical pair. As in Matolcsi and Ruzsa [5], triangulations
of B have a crucial role in our paper. By a triangulation T of B, we mean a triangulation
of [B] where the set of vertices of T is B. In addition, T is called stacked if it has B−d
fulldimensional simplices (which is the minimum possible number of simplices in a
triangulation of B points in Rd ). As first steps in the characterization, we show that
for every kcritical pair ( A, B):
• B ⊂ A (Lemma 18).
• ( A ∩ [B ], B ) is also kcritical, for any subset B ⊂ B (Lemma 19).
• B is totally stackable (Corollary 21), meaning that all of its triangulations are
stacked.
Total stackability is a very restrictive property that can be expressed in different ways
(Lemma 11) and totally stackable sets are completely characterized by Nill and Padrol
(see Theorem 12). Section 3 includes these results and some preliminary background
on triangulations. The fact that B needs to be totally stackable in order to have equality
follows from the following refinement of Theorem 1 that we prove in Sect. 6.
Theorem 3 Let T be a shellable triangulation of B with hvector (h0, . . . , hd ). Let
A be such that B ⊂ A ⊂ [B]. Then,
d + k
k
The hvector (h0, h1, . . . , hd ) ∈ Nd+1 (here and in what follows N = {0, 1, 2, . . . })
of a ddimensional triangulation is a classical invariant in geometric combinatorics,
which can be read either from the f vector (the number of simplices of each dimension)
or from a shelling. See more background on this topic in Sect. 3. Since hi ≥ 0 for every
i , Theorem 3 implies Theorem 1. But it also tells us that in order to have equality in
Theorem 1 all the shellable triangulations of B need to have hi = 0 for all i ≥ 2, which
is equivalent to them having B − d simplices. Hence, B needs to be totally stackable.
It is worth noticing that the inequality in Theorem 3 is equivalent to
For the case A = B this leads to the following refinement of Corollary 2:
Corollary 4 Let T be a shellable triangulation of A with hvector (h0, . . . , hd ). Then,
The geometric structure of critical pairs is complemented by its arithmetic structure.
To express this arithmetic structure we introduce the following concepts. For finite
B ⊂ Rd , we write (B) to denote the additive subgroup of Rd generated by B − B,
and hence by B if 0 ∈ B. We note that (B) is called a lattice if it is of rank d, which
will be the typical case.
Definition We say that A ⊂ [B] is stable with respect to B, or Bstable if
( A +
(B)) ∩ [B] = A.
The fact that A is Bstable provides a substantial arithmetic structure to A. For
example, suppose that A is Bstable and let l be a line intersecting A and such that
(B) contains nonzero vectors parallel to l. Let w be the shortest such vector (which
is unique up to sign). Then A∩ l can be partitioned into arithmetic progressions with
common difference w, each of which equals (x + Zw) ∩ [B] for some x ∈ Rd . If, in
addition, l contains an edge [u, v] of [B], then one of these arithmetic progressions
contains the vertices u, v of the edge. In particular, for two parallel lines l, l
intersecting A these arithmetic progressions have the same common difference (w depends
only on the direction of l) and if the lines contain edges e, e with (e) ≥ (e ) of B
then the translation of A∩ e within e matching one vertex of e is contained in A∩ e.
As usual, a ddimensional (straight) prism P is the Minkowski sum of a
(d − 1)dimensional polytope Q and a segment s not parallel to it. Edges of P
parallel to s are called vertical. We are interested in the case where Q is a simplex and
where the vertical edges are allowed to have different lengths. We abbreviate this as a
simplexprism. Put differently, a simplex prism is any polytope afffinely equivalent to
(x , t ) ∈ Rd+1 : x ∈
⊂ Rd , t ∈ R, (x ) ≤ t ≤ u(x ) ,
where
x ∈
is a dsimplex and , u are two affine functions with (x ) < u(x ) for every
.
With these geometric and arithmetic ingredients, Sects. 7, 8 and 9 lead to the
following explicit characterization of the critical pairs via a case study based in the
characterization of totally stackable sets.
Theorem 5 Let k ≥ 1, d ≥ 1, and let A, B ⊂ Rd be finite such that B spans Rd , and
A ⊂ [B]. Equality holds in Theorem 1 if and only if B ⊂ A, B is contained in the
union of the edges of [B], and one of the following conditions hold.
(i) B = d + 1. That is, B is the vertex set of a dsimplex.
(ii) For d ≥ 1, B consists of the vertices of the simplex [v0, . . . , vd ], and some extra
points on the edge [v0, vd ]. The points of B on this edge are part of an arithmetic
progression D contained in A, and A\(B ∪ D) is the disjoint union of translates
of D \ {v0}.
(iii) For d ≥ 2, [B] is a simplexprism, A is stable with respect to B, and A is
contained in the vertical edges of B.
(iv) For d = 2, A consists of the vertices of a triangle and the midpoints of its sides.
(v) For d = 2, B consists of the the vertices of a parallelogram, and A is stable with
respect to B and contained in the boundary of [B].
(vi) For 2 ≤ q < d, A and B are the unions of some d − q points and sets A , B
respectively, where ( A , B ) is a pair of qdimensional sets of type (iii), (iv) or
(v). That is to say, [ A] = [B] is an iterated pyramid over [ A ] = [B ] and the
only points of A \ A are the new vertices.
Observe that in part (vi) we do not include (iterated) pyramids over the
configurations of parts (i) and (ii) because these are again configurations of the types described
in (i) and (ii).
The characterization in Theorem 5 reveals two interesting facts about critical pairs.
• The characterization is independent of k. One direction (the fact that kcriticality
implies (k − 1)criticality, if k ≥ 2) is proved in Lemma 22. The other direction is
only proved as a consequence of the full characterization.
• If ( A, B) is critical then A is stable with respect to B. Actually, criticality of the
pair ( A, B) depends on A and the lattice (B) generated by the points of B rather
than the structure of B itself. Again, without resorting to the full characterization,
we only have a partial direct proof of this, namely the case of dimension one
(Proposition 8).
In turn, Theorem 5 yields the following concerning the equality case of Corollary 2.
Corollary 6 Let k ≥ 2, d ≥ 2, and let the finite A span Rd . Equality holds in
Corollary 2 if and only if one of the following conditions hold.
(i) The set A consists of the vertices of a simplex plus an arithmetic progression
contained in an edge of the simplex, starting and ending at the endpoints of the
edge.
(ii) The set A consists of the vertices of a simplex plus the midpoints of the sides of
a certain 2face of the simplex.
(iii) For d ≥ q ≥ 2, [ A] is an iterated pyramid over a qdimensional simplexprism.
There exists a nonzero w ∈ Rd such that A consists of the vertices of [ A] and,
for each vertical edge of the prism, the arithmetic progression of difference w
starting and ending at its endpoints.
Actually, Corollary 6 admits the more concise form of Corollary 7. In it, we say
that a triangulation T of a finite set A spanning Rd is unimodular if ( A) is a lattice
with determinant , and each full dimensional simplex of T has volume /d!. We
note that if A has a stacked unimodular triangulation then all of its triangulations are
unimodular and stacked.
Corollary 7 Let k ≥ 2, d ≥ 2, and let the finite A span Rd . Equality holds in
Corollary 2 if and only if A has a stacked unimodular triangulation.
To prove Theorem 5, first we consider the onedimensional case in Sect. 2, which
is the base of the arithmetic structure of critical pairs. Next we discuss some useful
properties of triangulations of convex polytopes in Sect. 3. Section 4 reviews the proof
of the Matolcsi–Ruzsa inequality Theorem 1, and concludes with a technical, but
useful, characterization (Theorem 15) of the equality case. Based on this result, we
show in Sect. 5 that the pairs ( A, B) listed in Theorem 5 are kcritical for any k ≥ 1.
Theorem 15 is also the base of the arguments leading to the fundamental properties
of kcritical pairs in Sect. 6. Finally, a case by case analysis in Sects. 7, 8 and 9
describes explicitly the arithmetic structure of the cases in Theorem 5. In Sect. 10
we show how the results of the previous sections imply that the list in Theorem 1 is
complete.
2 The Case of Dimension One
It is instructive to discuss the onedimensional version of Theorem 5 first, because it
does not require the geometric machinery built later on, and it provides the base of the
arithmetic structure of higher dimensional critical pairs.
For rational 0 ≤ b1 < · · · < bn, n ≥ 2, we define gcd{b1, . . . , bn} to be the
largest rational number w such that b1/w, . . . , bn /w are integers. We observe that if
A, B ⊂ R are finite such that A ⊂ [B] = [0, 1], then A being stable with respect
to B is equivalent to saying that B ⊂ Q, and A is the union of maximal arithmetic
progressions in [0, 1] with difference w = gcd(B).
We note that the onedimensional version of Theorem 1 reads as follows. If
A, B ⊂ R are finite sets with A ⊂ [B], and k ≥ 1, then
 A + k B ≥ (k + 1)( A − 1) + 1.
The first part of the next proposition gives the onedimensional version of
Theorem 5. The second part will be used later.
Proposition 8 Let k ≥ 1, and let A, B ⊂ R be finite such that A ⊂ [B] = [0, 1].
(i) The pair ( A, B) is kcritical if and only if {0, 1} ⊂ A and A is stable with respect
to B.
(ii) If C ⊂ (
0, 1
) is finite, then
C + k B ≥ (k + 1)C ,
with equality if and only if C is stable with respect to B.
Proof If a finite D ⊂ [B] is stable with respect to B, then
D + k B = D + k{0, 1},
and hence equality holds in (
3
) for D = C , and also in (
2
) for D = A provided that
{0, 1} ⊂ A.
We note that if either 0 ∈ A or 1 ∈ A, then a translate of A is contained in (
0, 1
),
a case dealt with in (ii), which shows that ( A, B) is not kcritical. Thus let the pair
( A, B) be kcritical with {0, 1} ⊂ A.
If B = {0, 1}, then A is clearly stable with respect to B. Therefore we may assume
that B ≥ 3. We write X to denote the image of X ⊂ R in the torus R/Z by the
quotient map. In particular
 A  =  A − 1.
Let A ⊂ [0, 2) be the set obtained by choosing the smallest element of A + B in each
coset of Z intersecting A + B. Since 0 ∈ B yields that A ∩ ( A + 1) = ∅, the sum
A + k B contains the disjoint union
{k + 1} ∪ A ∪ (( A\{1}) + 1) ∪ · · · ∪ (( A\{1}) + k).
We deduce using  A + B  ≥  A  that
 A + k B ≥  A + B  + k A  + 1 ≥ (k + 1) A  − 1.
(
5
)
(
2
)
(
3
)
(
4
)
As the pair ( A, B) is kcritical, (
2
) and (
4
) yield that  A + B  =  A . In particular
a + b ∈ A
for a ∈ A and b ∈ B .
We deduce from B ≥ 3 that there exists some nonzero element of B , which in turn
implies by the finiteness of A that B generates a finite subgroup H of R/Z, and A
is the union of some cosets of H . It follows that B ⊂ Q, and H is generated by w
for w = gcd(B). This implies (i). The argument for (ii) is completely analogous, only
k + 1 ∈ C + k B, and hence (
5
) is replaced by C + k B ≥ C + B  + kC  where
C  = C .
3 Some Observations about Triangulations
Throughout this paper, a triangulation of a finite point set B ⊂ Rd is a geometric
simplicial complex with vertex set B and underlying space [B]. A triangulation will
be given as a list of dsimplices.
Let T = {S1, . . . , Sm } be a triangulation of B. We say that the ordering S1, . . . , Sm
of the simplices of T is a shelling if, for every i , the intersection of Si with S1∪· · ·∪Si−1
is a union of facets of Si . Equivalently, if S1 ∪ · · · ∪ Si is a topological ball for every i .
The index of a simplex Si in a shelling is the number of facets of Si that are contained
in S1 ∪ · · · ∪ Si−1. That is, the index of S1 is zero and the index of every other Si is
an integer between 1 and d. The hvector of a shelling is the vector h = (h0, . . . , hd )
with hi equal to the number of simplices of index i . We recall without proof some
simple facts about shellings and hvectors (see [4, Sect. 9.5.2] or [10, Chap. 8] for
details):
Lemma 9 (i) Not every triangulation is shellable, but every point set has shellable
triangulations. For example, all regular triangulations (which include placing,
pulling and Delaunay triangulations) are shellable.
(ii) The hvector of a shellable triangulation is independent of the choice of shelling.
In fact, the hvector of a (perhaps nonshellable) triangulation can be defined as
hk =
k
i=0
(−1)k−i d + 1 − i
k − i
fi−1,
where ( f−1, . . . , fd ) is the f vector of T . That is, fi is the number of i simplices
in T , with the convention that f−1 = 1.
(iii) Every triangulation of B has h0 = 1, h1 = B − d − 1, and hi = m, where
m and d are the number of dsimplices and the dimension of T .
One useful way of constructing triangulations of a point set is the placing procedure,
which is recursively defined as follows (see [4, Sect. 4.3.1] for more details). Let
B ⊂ Rd be a finite point set and let x ∈ B be such that B := B \ {x } is ddimensional
and x ∈ [B ]. If T is a triangulation of B , we call placing of x in T the triangulation
T of B obtained adding to T the pyramids with apex at x of all the boundary (d − 1)
simplices of T that are visible from x . Here, we say that a (d − 1)simplex S in
the boundary of B is visible from x if its supporting hyperplane H separates x from
B \ H . Equivalently, if [x , y] ∩ [B ] = {y} for every point y ∈ S. It can be shown
that if T is shellable then T is shellable too.
The placing procedure can be used to construct a (shellable) triangulation of B from
scratch, by choosing an initial simplex S = [x1, . . . , xd+1] with {x1, . . . , xd+1} ⊂ B
and S ∩(B \{x1, . . . , xd+1}) = ∅, or to extend a given triangulation of a subset B ⊂ B
with [B ] ∩ (B \ B ) = ∅.
We observe that if C = {x1, . . . , xd+1} is affinely independent, s ∈ {1, . . . , d + 1},
and t > 0, then for the facets Fj = [C \x j ] of [C ], j = 1, . . . , s, we have
s
t · [C ]\ ∪ j=1 Fj
=
λ j x j : λ j > 0 for j ≤ s, ∀λ j ≥ 0,
λ j = t . (
6
)
d+1
j=1
d+1
j=1
Therefore if k ≥ 1, and S1, . . . , Sm is a shelling of a triangulation T , then
Ti + k Si = (k + 1)Ti for i = 2, . . . , m and Ti = Si \(S1 ∪ · · · ∪ Si−1).
(
7
)
Of special interest for us will be stacked triangulations. A stacked triangulation is
one that satisfies any of the following equivalent properties, and they are a particular
case of placing triangulations, hence shellable.
Lemma 10 The following properties are equivalent, for a triangulation T of a point
set B.
(i) The number of dsimplices in T equals B − d.
(ii) hi = 0 for all i ≥ 2.
(iii) The dual graph of T is a tree. The dual graph is the graph having as vertices the
dsimplices of T and as edges the adjacent pairs (pairs that share a facet).
(iv) Every simplex of dimension at most d − 2 of T is contained in ∂[B].
Proof The equivalence of the first two properties follows from hi = m and h0 +
h1 = B−d, where m denotes the number of dsimplices. For a shellable triangulation
T , the implications (ii)⇒(iii)⇒(iv)⇒(ii) are also trivial. Hence, the only thing we need
to prove is that any of (i), (iii) and (iv) implies T to be shellable. Let the simplices in T
be ordered S1, . . . , Sm in such a way that Si shares at least one facet with S1 ∪. . .∪ Si−1,
which can always be done. Then:
(i) S1 ∪ · · · ∪ Si has at most one vertex more than S1 ∪ · · · ∪ Si−1. If the total number
of vertices equals B − d we need the number to always increase by one, which
implies (S1 ∪ · · · ∪ Si−1) intersects Si only in a facet.
(iii) If the dual graph is a tree, it has one less edge than vertices. Then, no Si has two
facets in common with S1 ∪ · · · ∪ Si−1. It may in principle have a facet plus some
lower dimensional face σ , but this would imply the dual graph of the link of σ
in S1 ∪ · · · ∪ Si to become disconnected. Since at the end of the process all links
have connected dual graphs, there has to be a j > i such that S j also contains σ
and is glued to S1 ∪ · · · ∪ S j−1 along at least two facets, a contradiction.
(iv) If every simplex of dimension at most d − 2 of T is contained in ∂ [ B], then every
(d − 1)simplex in T disconnects T . Hence the dual graph is a tree and, by the
previous argument, T is shellable.
We call a point set B totally stackable if all its triangulations are stacked. This poses
heavy restrictions on the combinatorics of B, as we now see:
Lemma 11 Let B ⊂ Rd be a ddimensional finite point set. The following conditions
are equivalent:
(i) B is totally stackable.
(ii) Every k points of B lie in a face of [ B] of dimension at most k, for every k.
(iii) Every subset C of at most d − 1 points of B has [C ] ⊂ ∂ [ B].
Proof The implication (ii)⇒(iii) is obvious, and (iii) clearly implies the last property
of Lemma 10 for every triangulation, hence it implies (i). So, we only need to show
(i)⇒(ii).
Let C ⊂ B be a set of k points and let F be the minimal face of [ B] containing
C (the carrier of C ). Assume that dim(C ) > k and, without loss of generality, that
C is affinely independent. It is easy to show that BF := B ∩ F has a triangulation
TF using C as a simplex. Since [C ] goes through the interior of F , the link of C in
TF is a (dim(C ) − k)sphere. In particular, since dim(C ) − k > 0, its dual graph has
cycles. This TF can be extended to a triangulation of B (for example via the placing
procedure, see [4, Section 4.3.1]) which will still have cycles in its dual graph.
Properties (ii) and (iii) have the following straightforward consequences, which
will be useful in order to give an explicit description of all possible totally stackable
sets:
• If B is totally stackable, every point of B is either a vertex of [ B] or lies in the
relative interior of an edge of [ B]. That is, B is contained in the union of edges of
[ B]. We call the edges of [ B] that contain points of B other than vertices loaded.
• Every subset B of a totally stackable set B is totally stackable in aff ( B ).
Sets satisfying property (iii) of Lemma 11 are called of combinatorial degree one
by Nill and Padrol [3], who give a complete classification of them. The
description uses iterated pyramids, which we define in terms of the join operator. Let B1
and B2 be two finite sets in Rd whose affine hulls are of dimensions d1 and d2,
respectively. We say that B1 ∪ B2 is a join of B1 and B2 if the affine hull of
B1 ∪ B2 is of dimension d1 + d2 + 1. For i = 1, 2, consider a triangulation for
Bi where the number of di simplices is mi . These two triangulations induce a
triangulation for the join B1 ∪ B2 where the number of (d1 + d2 + 1)simplices is
m1m2. Moreover, all triangulations of a join arise in this way. A join where B2 is a
single point is a pyramid, and if B2 is affinely independent it is an iterated pyramid
over B1.
Theorem 12 (Nill and Padrol [3]) Let B be a finite set in Rd not contained in a
hyperplane. Then B is totally stackable if and only if B is contained in the union of
the edges of [ B], and either of the following conditions holds.
(i) [B] is a simplex, and all loaded edges meet at a vertex.
(ii) [B] is an iterated pyramid over a polygon, and every loaded edge is a side of the
polygon.
(iii) [B] is (projectively equivalent to) an iterated pyramid over a simplexprism, and
every loaded edge is a vertical edge of the prism.
Observe that if the polygon in case (ii) is a triangle then [B] is a simplex (as in case
(i)), but still the two cases differ in which edges are allowed to be loaded. In part (iii) we
need to allow projective equivalence on the prisms since being stackable is invariant
under it. The effect of a projective deformation on a prism is that the “vertical” edges
may no longer be parallel, but rather span lines meeting at a point.
4 A Proof of Theorem 1 and Some Consequences for Critical Pairs
In this section, we review the proof of Theorem 1 from Matolcsi and Ruzsa [5] in
order to analyze the equality case. This will lead to a technical but useful
characterization of critical pairs (Theorem 15), a strengthening of the Matolcsi–Ruzsa inequality
(Theorem 20), and various fundamental properties of critical pairs (Theorem 23).
Recall that k M = M + · · · + M denotes the kfold Minkowski sum of M with itself
and k M˙ = {k x : x ∈ M } denotes the dilation of M by a factor of k. We note that if
M is a convex set in Rd , and k ≥ 1 is an integer, then
k M = {k x : x ∈ M } = k · M.
(
8
)
The following simple observation will be often used.
Claim 13 Let S = [C ] be a dsimplex for C = {v0, . . . , vd }.
(i) kC  = d +kk .
(ii) For distinct points a, b ∈ S and k ≥ 1 we have
(a + kC ) ∩ (b + kC ) = ∅,
unless both a, b ∈ C .
Proof We may assume that v0 is the origin, and v1, . . . , vd form the orthonormal basis.
In this case we have
kC =
(t1, . . . , td ) ∈ Nd :
ti ≤ k ,
d
i=0
and hence (i) follows by enumeration.
For (ii), we observe that, for each pair x , y ∈ kC of distinct points, the sets x +
[0, 1)d and y + [0, 1)d are disjoint. Since S\{v0, . . . , vd } ⊂ [0, 1)d , it follows from
(a + kC ) ∩ (b + kC ) = ∅ that either a or b is a vertex, say a is a vertex of S. Then
x + a ∈ Zd , thus b is a vertex of S as well.
Corollary 14 If C = {v0,...,vd} is the vertex set of a simplex and if A ⊂ [C], then
A + kC = d +k k A −
A∩C d + k
i=1
k −
A + kC ≥ d +k k A − k k + 1
d + k ,
with equality if and only if C ⊂ A.
Proof Clearly,
A + kC = (a + kC).
a∈A
By Claim 13(i) each of the sets a + kC has cardinality d+k and, by Claim 13(ii),
k
they are pairwise disjoint except when a,a ∈ C ∩ A. That is:
A + kC = d +k k A \ C + (A ∩ C) + kC.
To prove (i) we only need to check that
(A ∩ C) + kC =
i=1
A∩C d + k + 1 − i .
k
For this assume, as in the proof of Claim 13(i), that v0 is the origin, and v1,...,vd
form an orthonormal basis. Let Cl := {v0,v1,...,vl−1}, 1 ≤ l ≤ d + 1, and assume
that A ∩ C = Ct for some t. Observe that
C1 + kC = kC,
and, for each l = 2,...,d + 1,
Hence,
id=0 ti = k + 1
(Cl + kC) \ (Cl−1 + kC) = (t1,...,td) ∈ Nd : t1 = ··· = tl−2 = 0 .
tl−1 > 0
(Cl + kC) \ (Cl−1 + kC) =
d + k + 1 − l ,
k
and
( A ∩ C ) + kC  = Ct + kC  =
i=1
A∩C d + k + 1 − i
.
k
For the second part of the statement, observe that in
each summand is nonnegative, so
A∩C d+k
i=1 k
−
,
with equality if and only if  A ∩ C  = d + 1.
Proof of Theorem 1 Let S1, . . . , Sm be a shelling of a triangulation T of B. Let Ci be
the set of vertices Si . According to (
8
),
(k + 1)Si , i = 1, . . . , m, form a triangulation of (k + 1)[B].
(
9
)
We define
A1 = A ∩ S1,
Ai = A ∩ Si \(S1 ∪ · · · ∪ Si−1)
for i = 2, . . . , m.
We observe that A1, . . . , Am form a partition of A. Moreover, by shellability,
Ai ⊂ Si , and Ai contains at most one vertex of Si for i = 2, . . . , m.
We deduce from (
7
) that
 A + k B ≥
 Ai + kCi .
m
i=1
(with equality if and only if C1 ⊂ A1), and Claim 13 (i) and (ii) imply by (
10
) that
Theorem 1 follows from combining (11), (12) and (13).
Using the notation of the above proof, the following characterization of equality in
Theorem 1 follows from (
7
) and (
9
) on the one hand, and (11), (12) and (13) on the
other hand.
Theorem 15 Let A, B ⊂ Rd finite such that A ⊂ [B] and dim[B] = d, and let k ≥ 1.
The pair ( A, B) is kcritical if and only if for some shelling S1, . . . , Sm of an arbitrary
triangulation T of B, we have
(i) C1 ⊂ A;
(ii) Ai + kCi = ( A + k B) ∩ (k + 1)Ti for i = 1, . . . , m
where Ci denotes the set of vertices Si , T1 = S1 and Ti = Si \(S1 ∪ · · · ∪ Si−1), i ≥ 2,
and Ai = A ∩ Ti for i = 1, . . . , m.
We will also use the following consequence of the proof of Theorem 1.
Lemma 16 Suppose that A + x ⊂ int[B] for some x ∈ Rd . Then
(14)
Proof If A + x ⊂ int[B] for some x ∈ Rd , then A ∩ B = ∅ can be assumed, and hence
Corollary 14 gives, in the notation of the proof of Theorem 1,  A1 + kC1 ≥ d +kk  A1.
Therefore (11) and (13) yields (14).
5 Proof of Sufficiency in Theorem 5
Based on Theorem 15, we show that the pairs ( A, B) in Theorem 5 are kcritical for
any k ≥ 1. First we show that we can restrict B to the vertices of [B] in the case of
the pairs ( A, B) listed in Theorem 5.
Lemma 17 If ( A, B) is any of the pairs listed in Theorem 5, and B is the vertex set
of [B], then
A + k B = A + k B for any k ≥ 1.
Proof It is sufficient to prove that for any a ∈ A and b ∈ B, there exist a ∈ A and
b ∈ B such that a + b = a + b . Since B ⊂ A, we may assume that a, b ∈ B . In
particular, we may assume that one of the cases (ii)–(iv) holds.
Write {a, b} = {a˜ , b˜}, where a = a˜ and b = b˜ in the cases (ii) and (iv), and
vertical edge of [B] containing b˜ is not longer than the vertical edge containing a˜ in
the case (iii). The fact that A is stable with respect to B and the conditions in (ii)–(iv)
of Theorem 5 mean that b˜ belongs to an arithmetic progression D along an edge [u, v]
of [B] containing its two vertices u, v, and a˜ belongs to x + D\{u} ⊂ A for some x .
The result follows since D + D\{u} = {u, v} + D\{u}.
Proof of Sufficiency in Theorem 5 Let us assume that the pair ( A, B) satisfies one of
the conditions (i)–(vi) in Theorem 5.
If [B] is a simplex, then combining Lemma 17 and Corollary 14 yields equality
in Theorem 1. This covers the cases (i), (ii) and (iv) of Theorem 5, and the part of
case (vi) when the pair ( A, B) is obtained by adding d − 2 points to a pair ( A , B )
described in the case (iv).
Therefore we assume that [B] is an iterated pyramid over a qdimensional
simplexprism P, 2 ≤ q ≤ d, and referring to Lemma 17, also that B consists of the vertices of
[B]. Let B0 = B\(B ∩ P). We write v1, . . . , vq , w1, . . . , wq to denote the vertices of
P in a way such that the vectors wi − vi are parallel pointing into the same direction
for i = 1, . . . , q. We define Si = [{v1, . . . , vi , wi . . . wq } ∪ B0] for i = 1, . . . , q, and
hence S1, . . . , Sq form a shelling of the corresponding triangulation of B. We write
Ai , Ci , Ti to denote the corresponding sets defined in Theorem 15 for i = 1, . . . , q.
Let k ≥ 1 and i ∈ {1, . . . , q}. We claim that assuming vi = 0, we have
A + k B = ( Ai +
We distinguish two cases. If P is parallelogram, then P is actually a fundamental
parallelogram for the twolattice (B) ∩ lin P. Since A is stable with respect to (B),
we deduce (15) by (17). In addition (16) follows from (
7
), and the fact that the nonzero
elements of Ci form a Zbasis of (B). This covers the case (v) of Theorem 5, and
the corresponding part of the case (vi).
Finally we consider the case (iii) of Theorem 5, and the corresponding part of the
case (vi). We assume that P is not a parallelogram, and hence A ∩ P is contained in
the vertical edges [v j , w j ] of P, j = 1, . . . , q. Let
{z1, . . . , zd−1} = B0 ∪ ({v1, . . . , vq }\{vi }),
thus there exists w ∈ Rd pointing into the same direction as wi − vi such that
{w, z1, . . . , zd−1} form a Zbasis for (B). It follows that there exist integers
m1, . . . , mq ≥ 1 such that w j − v j = m j w for j = 1, . . . , q, and there exists
⊂ [0, 1) such that v j + t w ∈ A for j ∈ {1, . . . , q} and t ∈ [0, m j ] if and only
if t − t ∈ . We define the integers n1, . . . , nd−1 by n p = m j if z p = m j , and
n p = 0 if z p ∈ B0. Writing
= {(i1, . . . , id−1) ∈ Zd−1 : i1 + · · · + id−1 = k and i j ≥ 0},
we deduce (15) from (17) and
d−1
A + k B =
t w +
i j z j : t − t ∈
and 0 ≤ t ≤
= ( Ai +
Turning to (16), we observe that Ci \{vi } form a basis for Rd . Combining this fact
with (
7
) yields
Since {w} ∪ (Ci \{vi , wi }) form a Zbasis for (B), and Ai + w ⊂ Ai + (Ci ), we
deduce that Ai + (Ci ) = Ai + (B). We conclude (16), and in turn that equality
holds in Theorem 1 for the pairs ( A, B) in Theorem 5.
6 Basic Properties of kCritical Pairs
The goal of the section is to prove Theorem 23 listing some fundamental properties
of kcritical pairs. The first one is a direct consequence of Theorem 15.
Lemma 18 If ( A, B) is a kcritical pair, k ≥ 1, then B ⊂ A.
Proof For any x ∈ B, we consider a shellable triangulation T with x ∈ C1 for the first
simplex S1 = [C1] of T (this can be achieved, for example, via a placing triangulation).
Theorem 15 yields C1 ⊂ A, thus x ∈ A.
Based on Theorem 15, we prove that criticality is preserved by taking subsets of B.
Lemma 19 Let A, B be ddimensional point sets with A ⊂ [B], and let k ≥ 1. If
( A, B) is kcritical, then ( A ∩ [B ], B ) is also kcritical for every B ⊂ B.
Proof Let B = B ∩ [B ] and A = A ∩ [B ]. Since B ⊂ B and [B ] = [B], it is
sufficient to prove that ( A, B) is kcritical.
If dim [B ] = d, then constructing a placing triangulation first for B, we obtain
some shelling S1, . . . , Sm of a triangulation of B such that the union of S1, . . . , Sn is
[B ] for some n ≤ m. Now Theorem 15 (i) and (ii) for the pair ( A, B) readily yield
the analogous properties for the pair ( A, B).
Next we assume that dim[B ] = q < d. We choose x1, . . . , xd−q ∈ B such that
for B∗ = {x1, . . . , xd−q } ∪ B, we have dim[B∗] = d and B ∩ [B∗] = B∗. Let
A∗ = A ∩ [B∗]. We observe that L = aff B is a supporting qplane to [B∗], with
A = A∗ ∩ L, B = B∗ ∩ L and [B] = [B∗] ∩ L.
Let S1, . . . , Sn be a shelling of some triangulation of B, and let Ai , Ci , Ti for
i = 1, . . . , n be the corresponding sets for Theorem 15. We need to prove that they
satisfy Theorem 15. Since B ⊂ A according to Lemma 18, Theorem 15 (i) readily
follows, and all we have to verify is Theorem 15 (ii).
To achieve that, we observe that S1, . . . , Sn is a shelling of a triangulation of B∗
where Si = [x1, . . . , xd−q , Si ] for i = 1, . . . , n. Writing Ai , Ci and Ti to denote the
corresponding sets in Theorem 15, we have A1 = A1 ∪ {x1, . . . , xd−q }, Ai = Ai for
i = 2, . . . , n, moreover Ci = Ci ∩ L and Ti = Ti ∩ L for i = 1, . . . , n. The pair
( A∗, B∗) is kcritical by the argument above because B∗ ⊂ B with dim B∗ = d, and
hence Ai , Ci and Ti satisfy Theorem 15 (ii). It follows that the same conclusion holds
for Ai , Ci and Ti , as for i = 1, . . . , n, we have
( A + k B) ∩ Ti = L ∩ ( A∗ + k B∗) ∩ Ti ⊂ L ∩ ( Ai + kCi ) = Ai + kCi ,
where the first and the last equality is a consequence of the fact that L is a supporting
qplane to [B∗].
Under the assumption B ⊂ A, the ideas in the proof of Theorem 1 also lead to a
proof of Theorem 3. We recall its statement for the convenience of the reader.
Theorem 20 Let A, B ⊂ Rd be finite such that dim [B] = d and B ⊂ A ⊂ [B], and
let T be a shellable triangulation of B with hvector (h0, . . . , hd ). Then
h j
Proof Let S1, . . . , Sm be a shelling of T . We keep the same notation for Ai , Ci and
Ti as in the statement of Theorem 15. In particular, we have
m
i=1
( A + k B) ∩ (k + 1)Ti  ≥  Ai + kCi  =  Ai + kCi  + (k + 1)Ci ∩ (k + 1)Ti . (18)
The first summand is
by Claim 13 (i), and so
m
i=1
For the second summand, let si be the index of Si in the shelling, and hence (
6
) and
enumeration yield
d+k+1−si
k+1−si
if si ≤ k + 1,
otherwise.
Put differently,
h j
Since (k + 1)Ci and Ai + kCi are disjoint by Claim 13 (ii), we obtain
h j
j=0
d + k + 1
k + 1
h j
h j
where, in the last step, we use h0 = 1 and h1 = B − d − 1.
Corollary 21 If the pair ( A, B) is kcritical with dim[B] = d, then B is totally
stackable. In particular, B is contained in the union of the edges of [B].
Proof We have B ⊂ A according to Lemma 18. Theorem 20 yields that every shellable
(in particular, every regular) triangulation has h2 = 0. According to the
characterization of the hvectors by Stanley [8] (see also Theorem 8.34 in Ziegler [10]), we
have h j = 0 for j ≥ 2, which, by Lemma 10, implies that B is stacked. That is,
every regular triangulation of B has B − d dsimplices. It is a fact (see [4, Theorem
8.5.19]) that then all the triangulations (regular or not) have the same number B − d
of dsimplices. That is, B is totally stackable.
Next we prove that equality in Theorem 1 is preserved under reducing the value
of k.
Lemma 22 If ( A, B) is kcritical for k ≥ 2 with dim[B] = d, then it is also k critical
for every k = 1, . . . , k − 1.
Proof Let S1, . . . , Sm be a shelling of a triangulation of B. We use the notation of
Theorem 15. Condition (i) of Theorem 15 is independent of k, and hence we need to
check condition (ii).
Let z ∈ ( A + (k − 1)B) ∩ Ti for i = 1, . . . , m, and what we need to show is
that z ∈ Ai + (k − 1)Ci . Since B is totally stackable by Corollary 21, we may
assume that Ci = {v0, . . . , vd } in a way such that Ti = Si \[v1, . . . , vd ] if i ≥ 2, and
z ∈ [v1, . . . , vd ] if i = 1. In particular, v0 ∈ Ai , and
λ j v j where λ0 > 0, λ j ≥ 0 for j = 1, . . . , d, and λ0 + · · · + λd = k
by (
6
). If z ∈ kCi , or equivalently each λi is an integer, then v0 ∈ Ai and λ0 > 0 yield
that z ∈ Ai + (k − 1)Ci . Therefore we assume that z ∈ kCi .
Since z + v0 ∈ ( A + k B) ∩ (k + 1) Ti and ( A, B) is kcritical, Theorem 15 (ii)
yields that
z + v0 = a +
m j v j
(19)
where a ∈ Ai , every m j ≥ 0 is an integer. and m0 + · · · + md = k. We have
a =
α j v j where α0 > 0, α j ≥ 0 for j = 1, . . . , d, and α0 + · · · + αd = 1,
and α0 > 0 is a consequence of (
6
). As Ci is affinely independent, the coefficients
satisfy λ0 + 1 = α0 + m0 and λ j = α j + m j for j = 1, . . . , d.
Since z ∈ kCi , we deduce that some α j is not integer, which in turn implies that
α0 < 1. We have α0 + m0 = λ0 + 1 > 1 based on (19), thus m0 ≥ 1 by α0 < 1.
Therefore
z =
d
z = a + (m0 − 1)v0 +
m j v j ∈ Ai + (k − 1)Ci ,
as it is required by Theorem 15 (ii).
We summarize Lemmas 18, 19 and 22 and Corollary 21 as follows.
Theorem 23 If the pair ( A, B) is kcritical for k ≥ 2 with dim [B] = d, then
(i) B ⊂ A;
(ii) ( A ∩ [B ], B ) is also kcritical for every B ⊂ B;
(iii) B is totally stackable, thus B is contained in the union of the edges of [B];
(iv) ( A, B) is 1critical, and hence  A + B = (d + 1) A − d(d + 1)/2.
From now on we consider kcritical pairs ( A, B) for k = 1, which will be simply
called critical pairs. Theorem 23 (iv) shows that kcritical pairs are critical. We also
speak about critical sets in the case of the one dimensional version  A + B ≥ 2 A − 1
of the Matolcsi–Ruzsa inequality.
7 The Case of a Simplex
(20)
(21)
In this section we consider the case where [B] is a dsimplex. First we discuss iterated
pyramids, a case that will be used later on as well.
Lemma 24 Let 1 ≤ q < d, and let ( A, B) be a critical pair with dim[B] = d such
that [B] is an iterated pyramid over [B0] where B0 ⊂ B and dim[B0] = q. Then
(i) ( A ∩ [B0], B0) is a critical pair, and
(ii) ( A ∩ L)+ B0 = (q +1) A ∩ L for any affine qplane L parallel to L0 = aff(B0)
intersecting A and avoiding the vertices of [B].
Proof We have B ⊂ A by Theorem 23 (i), and the pair ( A ∩ [B0], B0) is critical by
Theorem 23 (ii). Let [B] = [x1, . . . , xd−q , B0], and let B be the join of {x1, . . . , xd−q }
and B0. In particular, ( A, B) is a critical pair, by Lemma 19.
We may assume that 0 ∈ B0. We divide A into equivalence classes
according to the cosets of H = Zx1 + · · · + Zxd−q + L0, and hence adding B ⊂ H
to different equivalence classes results in disjoint sets. One equivalent class is
A = {x1, . . . , xd−q } ∪ ( A ∩ [B0]), and
 A + B ≥ (d + 1) A − d(d + 1)/2
by Theorem 1. Any other equivalence class is of the form A ∩ L for an affine qplane
L parallel to L0 that avoids B and intersects A. Since a translate of A ∩ L is contained
in the relative interior of [B0], Lemma 16 yields
( A ∩ L) + B ≥ (d + 1) A ∩ L.
As ( A, B) is a critical pair, (20) and (21) imply
d−q
i=1
(d + 1) A ∩ L = ( A ∩ L) + B = ( A ∩ L) + B0 +
( A ∩ L) + xi ,
therefore ( A ∩ L) + B0 = (q + 1) A ∩ L.
We recall that an edge of [B] is loaded if it contains at least three points of B.
Proposition 25 Let A, B be finite ddimensional sets in Rd , d ≥ 2 with A ⊂ [B]. If
[B] is a simplex, and ( A, B) is a critical pair, then B ⊂ A, B is contained in the edges
of [B], and one of the following conditions hold:
(i) B = d + 1; or
(ii) there is a unique loaded edge [u, v] of B; the points of B in this edge are part
of an arithmetic progression D contained in A; and A\(B ∪ D) is the disjoint
union of translates D \ {v}; or
(iii) there exist two or three loaded edges for B, which are sides of a two dimensional
face T of [B], and A consists of the vertices of [B], and the midpoints of the sides
of T .
A\(D ∪ B ) is the disjoint union of translates of D \ {v}.
(22)
If there exists a unique loaded edge of B, then (22) yields (ii). Therefore we may
assume that there are at least two loaded edges of [B]. Since B is totally stackable by
Theorem 23 (iii), it follows that either all loaded edges meet in a vertex, or they form
a triangular 2dimensional face by Theorem 12.
Therefore we may assume [v0, v1] and [v0, v2] are two loaded edges of B with
v0 = 0, and let T = [v0, v1, v2] be the 2face containing these two edges. In particular,
( A ∩ T , B ∩ T ) is a critical pair by Theorem 23 (ii). It follows by (22) that for i = 1, 2,
A ∩ T contains an arithmetic progression Di of length mi ≥ 3 with endpoints v0
and vi . According to (22), a1 = mm11−−21 v1 is part of a translate of D2\{v2} contained
in A ∩ T , and hence also of a segment σ ⊂ T of length at least mm22−−21 v2 . Since
mi −2
mi −1 ≥ 21 , we deduce that m1 = m2 = 3, D1 = {v0, a1, v1} and D2 = {v0, a2, v1}
for a2 = 21 v2. It follows by (22) that a0 = 21 (v1 + v2) = a1 + a2 ∈ A.
Let a ∈ A\(D1 ∪ B ), and let L = a + lin{v1, v2}. It follows by (22) applied to
the edge [v0, v1] that a ∈ { p, p − a1} where { p, p − a1} ⊂ A. Now p ∈ (D2 ∪ B ),
and hence applying (22) to the edge [v0, v2], we conclude that either p + a2 ∈ A, or
p−a2 ∈ A. In other words, either [ p, p−a1, p+a2] ⊂ [B]∩L, or [ p, p−a1, p−a2] ⊂
[B] ∩ L. Since [B] ∩ L is a translate of λT for λ ∈ (0, 1] and { p, p − a1} ∩ D1 = ∅,
we deduce that λ = 1, and { p, p − a1, p − a2} = {a0, a2, a1}. Therefore A consists
of the vertices of [B], and the midpoints of T .
8 Critical Pairs ( A, B) with dim[ B] = 2
In this section, A, B are finite sets in R2 satisfying that A ⊂ [B] and B spans R2.
Thus the case k = 1 of Theorem 1 can be written into the form
 A + B ≥ 3 A − 3.
(23)
We note that the case when [B] is a triangle is handled in Sect. 7, and hence we start
with the case when [B] is a quadrilateral.
Proposition 26 Let ( A, B) be a critical pair with [B] = [v0, v1, v2, v3] a
quadrilateral. Then A ⊂ ∂[B] and [B] is a trapezoid. Moreover
(i) if B consists of the vertices of a parallelogram, then A can be partitioned into
pairs of points, each a translate of a pair of consecutive points of B;
(ii) if B has a loaded edge or is not a parallelogram, then A is contained in two
parallel edges of B, say e1 = [v0, v1] and e2 = [v2, v3], and each of A∩ei can be
partitioned into maximal arithmetic progressions with common difference w; in
addition, if (e2) ≤ (e1) and [v0, v2] is an edge of [B], then ( A∩e2)−(v2−v0) =
A ∩ (e2 − (v2 − v0)).
Proof Let B = {v0, v1, v2, v3} be the vertices of B, where we may assume that v0 = o
and [0, v1], [0, v2] are edges of [B]. By Theorem 23, we may assume that B ⊂ A and
that ( A, B ) is critical.
Suppose that [B] has no pair of parallel sides. We say that a side of [B] is big if
the sum of the angles at the endpoints of the side is less than π . Since one side out of
two opposite sides of [B] is big, there exists a vertex of [B] where two big sides meet.
Therefore we may assume that B = {o, v1, v2, t1v1 + t2v2}, where
0 < t1 ≤ t2 < 1 and t1 + t2 > 1.
In addition, if s1v1 + s2v2 ∈ A then s1 + s2 ≤ t1 + t2.
Let B0 = {o, v1, v2}. We observe that any coset of Z2 = Zv1 + Zv2 intersects
[0, 1)v1 + [0, 1)v2 in exactly one point, therefore no two points of A\B0 are in the
same coset. We deduce that
 A + B0 =  A \ B0 · B0 + 2B0 = 3( A − 3) + 6 = 3 ·  A − 3.
On the other hand, if s1v1 + s2v2 ∈ A + B0 then s1 + s2 ≤ t1 + t2 + 1. Therefore
2(t1 + t2) > t1 + t2 + 1 yields
2(t1v1 + t2v2) ∈ ( A + B )\( A + B0),
and hence  A + B  >  A + B0 = 3 ·  A − 3, contradicting that ( A, B ) is a critical
pair. This proves the first part of the statement.
For (i), suppose that B = B and [B] is a parallelogram. We may assume that
[B] = [0, 1]2. We partition A = A0 ∪ · · · ∪ At into equivalence classes according to
Z2, where A0 = B. We observe that, for i > 0, each Ai consists either of one single
point or a pair which is a translate of a pair of consecutive vertices in B. We have
 A + B =
k
i=0
 Ai + B = 3 A0 − 3 +
 Ai + B = 3 A − 3,
k
i=1
which implies  Ai + B = 3 Ai  for each i = 1, . . . , k. This implies that no Ai consists
of a single point.
Finally, to prove (ii), we suppose that v1 = λ(v3 − v2) with λ ≥ 1 and that the edge
e1 = [o, v1] is loaded if λ = 1. Set Ai = A ∩ ei and Bi = B ∩ ei for i = 1, 2. Consider
equivalence classes of A determined by the cosets of the subgroup H = Zv2 + Rv1.
One equivalence class of A is A1 ∪ A2, and the rest are of the form A ∩ l for some
line l parallel to [o, v1], and intersecting int[B]. For such a line l we claim that
( A ∩ l) + B > 3 A ∩ l.
(24)
Indeed, if λ > 1 then
( A ∩ l) + B  = ( A ∩ l) ∪ (( A ∩ l) + v1) ∪ (( A ∩ l) + {v2, v3}) > 3 A ∩ l.
If λ = 1 then B1 > 2 by our assumption, and we have ( A ∩ l) + B1 ≥  A ∩ l +
B1 − 1 =  A ∩ l + 2. Hence,
( A ∩ l) + (B1 ∪ B2) = ( A ∩ l) + B1 + ( A ∩ l) + B2
≥ ( A ∩ l + 2) + (2 A ∩ l − 1) > 3 A ∩ l.
Since ( A1 ∪ A2) + (B1 ∪ B2) ≥ 3( A1 ∪ A2) − 3 by Theorem 1, it follows from
(24) that A = A1 ∪ A2, and in turn B ⊂ A implies that B = B1 ∪ B2. This shows that
both A and B are contained in two parallel lines of the trapezoid. Therefore,
 A + B =  A1 + B1 +  A2 + B2 + ( A1 + B2) ∪ ( A2 + B1),
where, by Proposition 8 (i),  Ai + Bi  ≥ 2 Ai −1 with equality if and only if Ai is stable
with respect to Bi . Moreover we have also ( A1 + B2) ∪ ( A2 + B1) ≥  A1 +  A2 − 1.
Hence, it follows from  A + B = 3 A−3 that there is equality in the three inequalities
above, which implies
A2 ⊆ A1 + v2 and ( A1 + v2) ∩ [v2, v3] ⊆ A2,
which together with the other two equalities imply (ii).
Lemma 27 If ( A, B) is a critical pair, and [B] is a polygon, then [B] has at most four
vertices.
Proof We suppose that P = [B] is a polygon of at least five vertices, and seek a
contradiction. According to Theorem 23, we may assume that P is a pentagon, and B
consists of the vertices of P. For any vertex v of P, let Pv be the convex hull of the
other four vertices of P. It follows again by Theorem 23 that ( A ∩ Pv, B ∩ Pv) is a
critical pair, as well, and hence Proposition 26 yields that Pv is a trapezoid.
Since the sum of the angles of P is 3π , there exists a side f of P such that the sum
of the angles at the two endpoints is at least 65π > π . Let e be the diagonal of P not
meeting f , and let v be the vertex not in e ∪ f . It follows that Pv is a trapezoid where e
and f are parallel, and (e) > ( f ). We deduce from Proposition 26 that there exists
x ∈ A ∩ e different from the endpoints of e.
Now let w be an endpoint of f . Since e is a diagonal of Pw, we have x ∈ A ∩ int Pw.
However Proposition 26 (i) and (ii) applied to the pair ( A ∩ Pw, B ∩ Pw) shows that
A ∩ int Pw = ∅, which is a contradiction.
9 Critical Pairs ( A, B) where [ B] is an Iterated Pyramid Over a SimplexPrism
Our first statement is a preparation for the proof of Lemma 29.
Lemma 28 If A, B ⊂ Rd , d ≥ 2, are finite such that x + A ⊂ int[B] for x ∈ Rd ,
and [B] is a ddimensional simplexprism, then
 A + B > (d + 1) A.
Proof Let [v0, . . . , vd−1] and [w0, . . . , wd−1] be the facets of [B] such that [vi , wi ]
are the parallel vertical edges for i = 0, . . . , d − 1, and
vi − wi ≤ v0 − w0 for i = 1, . . . , d − 1.
(25)
We may assume that v0 is the origin, and A ⊂ int[B]. Let B ={v0, w0, . . . , vd−1, wd−1}
be the vertex set of [B], and let B = {v1, w1, . . . , vd−1, wd−1}.
We divide A into equivalence classes according to the subgroup
H = Zv1 + · · · + Zvd−1 + Rw0,
and hence adding B ⊂ H to different equivalence classes results in disjoint sets. As
A ⊂ int[B] and (25) yield that A ⊂ (
0, 1
)v1 + · · · + (
0, 1
)vd−1 + (
0, 1
)w0, any
equivalence class is of the form A ∩ l for a line l parallel to w0 and intersecting A, and
a translate of A ∩ l is contained in [v0, w0]\{v0, w0}. Thus Proposition 8 (ii) implies
( A ∩ l) + {v0, w0} ≥ 2 A ∩ l.
We note that
therefore
the sets l + vi , i = 0, . . . , d − 1, are pairwise disjoint,
d−1
i=1
d−1
i=1
( A ∩ l) + B  =
( A ∩ l) + {vi , wi } ≥
( A ∩ l + 1) > (d − 1) A ∩ l. (28)
We conclude  A + B ≥  A + B > (d + 1) A by (26), (27) and (28).
Combining Lemmas 24 and 28 yield the following.
Lemma 29 If d > q ≥ 2 and ( A, B) is a critical pair such that B spans Rd , A ⊂
[B], and [B] = [x1, . . . , xd−q , P] for a qdimensional simplexprism P, then A =
{x1, . . . , xd−q } ∪ ( A ∩ P).
It remains to describe the structure of A and B when [B] is a simplexprism.
(26)
(27)
Proposition 30 If d ≥ 3, and ( A, B) is a critical pair such that [B] is a ddimensional
polytope projectively equivalent to a simplexprism, then B ⊂ A, and
(i) the vertical edges of [B] are parallel,
(ii) A is contained in the vertical edges of [B],
(iii) there exists a vertical vector w = 0 such that for each vertical edge e, A ∩ e
can be partitioned into maximal arithmetic progressions of difference w in e, one
of them containing both endpoints of e, and this longest arithmetic progression
contains B ∩ e. In addition if e and f are vertical edges, and e + v ⊂ f in a way
such that e + v and f share a common endpoint, then ( A ∩ e) + v = A ∩ (e + v).
Proof We have B ⊂ A by Theorem 23 (i). Let [v0, . . . , vd−1] and [w0, . . . , wd−1] be
the facets of [B] such that [vi , wi ] are the vertical edges for i = 0, . . . , d − 1.
For 0 ≤ i < j ≤ d − 1, it follows from Theorem 23 (iii) that ( Ai j , Bi j ) is a
critical pair for Ai j = [vi , wi , v j , w j ] ∩ A and Bi j = [vi , wi , v j , w j ] ∩ B, therefore
Proposition 26 yields that
[vi , wi , v j , w j ] is a trapezoid, and
Ai j and Bi j satisfy the conditions of Proposition 26 (i) or (ii).
We verify (i) using an indirect argument. We suppose that the lines of the vertical
edges meet at a point p ∈ Rd , and seek a contradiction. We may assume that wi ∈
[ p, vi ] for i = 0, . . . , d − 1. Since the pair composed of A ∩ [v0, v1, w1, v2, w2]
and B ∩ [v0, v1, w1, v2, w2] is critical by Theorem 23 (iii), and [v1, w1, v2, w2] is a
trapezoid according to (29) , Lemma 29 yields that
A ∩ [v0, v1] = {v0, v1}.
However wi ∈ [ p, vi ] for i = 0, 1 and (29) yield that [w0, w1] is parallel with
and shorter than [v0, v1]. We deduce from Proposition 26 that  A ∩ [v0, v1] ≥ 3,
contradicting (31), and implying (i).
We prove (ii) again by contradiction, therefore we suppose that there exists an
x ∈ A not contained in the vertical edges.
According to the Charateodory theorem (see, e.g., Grünbaum [2]), if x ∈ [X ]\X
for X ⊂ Rd , then x ∈ [x0, . . . , xd ] for x0, . . . , xd ∈ X . It follows that possibly after
reindexing, there exists m such that 1 ≤ m ≤ d, [x0, . . . , xm ] is an msimplex, and
x ∈ relint[x0, . . . , xm ] = {t0x0 + · · · + tm xm : t0, . . . , tm > 0 and t0 + · · · + tm = 1}.
(32)
We deduce from (32) that there exists 1 ≤ m ≤ d, and affinely independent vertices
x0, . . . , xm of [B] such that x ∈ relint[x0, . . . , xm ]. Since an msimplex has no two
parallel edges, we may assume that x0 = v0, and w0 ∈ {x1, . . . , xm }. In particular, x ∈
A ∩ P for P = [v0, v1, w1, . . . , vd−1, wd−1] where Q = [v1, w1, . . . , vd−1, wd−1]
is a simplexprism by (i). It follows from x ∈ relint[x0, . . . , xm ] that x = v0 and
x ∈ Q. Since the pair ( A ∩ P, B ∩ P) is critical, this contradicts Lemma 29, and hence
implies (ii).
The last property (iii) follows from (30) and Proposition 26.
(31)
10 Proof of Necessity in Theorem 5
Let ( A, B) be a kcritical pair for some k ≥ 1 with dim[ B] = d. In particular, ( A, B)
a 1critical and B is totally stackable by Theorem 23. According to Theorem 12, [ B]
is a simplex, or an iterated pyramid over a polygon, or over (a projective deformation
of) a simplexprism. If [ B] is a simplex, then the characterization in Theorem 5 (i),
(ii), (iv) and (vi) is achieved by Proposition 8 if d = 1, and Proposition 25 if d ≥ 2.
Therefore let [ B] be an iterated pyramid over P with dim P = q where P is polygon
or a projective deformation of a simplexprism. We may assume that P is not a triangle.
Since the pair ( A ∩ P, B ∩ P ) is 1critical by Theorem 23 (ii), Proposition 26 and 27
yield that if P is a polygon, then it is a trapezoid. In addition, Proposition 30 yields
that the vertical edges of P are parallel even if q ≥ 3.
We deduce from Lemma 29 that any point of A is a vertex of [ B], or contained in
P . Therefore we conclude Theorem 5 (iii) and (v) from Proposition 26 if q = 2, and
from Proposition 30 if q ≥ 3.
Acknowledgments Part of the research was done during an FP7 Marie Curie Fellowship of the first name
author at BarcelonaTech, whose hospitality is gratefully acknowledged. Károly J. Böröczky was supported
by the FP7 IEF Grant GEOSUMSETS and OTKA Grant 109789. Francisco Santos was supported by the
Spanish Ministry of Science (MICINN) through Grant MTM201122792. Oriol Serra was supported by
the Spanish Ministry of Science (MICINN) under Project MTM201128800C0201, and by the Catalan
Research Council under Grant 2009SGR1387.
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