On Sumsets and Convex Hull
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Abstract
One classical result of Freiman gives the optimal lower bound for the cardinality of \(A+A\) if \(A\) is a \(d\)dimensional finite set in \(\mathbb R^d\). Matolcsi and Ruzsa have recently generalized this lower bound to \(A+kB\) if \(B\) is \(d\)dimensional, and \(A\) is contained in the convex hull of \(B\). We characterize the equality case of the Matolcsi–Ruzsa bound. The argument is based partially on understanding triangulations of polytopes.
Keywords
Sumsets Shellable triangulations \(h\)vector1 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].
Theorem 1
In particular, taking \(A=B\) they get the following, of which (1) is the case \(k=2\):
Corollary 2
In these results, for a set \(X\subset \mathbb R^d\), we set \(1X=X\), \(kX=(k1)X+X\) for \(k\ge 2\), and \(0X=\{0\}\). The sum \(X+\emptyset \) is always the empty set. The convex hull of the set \(X\subset \mathbb R^d\) is denoted by \([X]\). Similarly, \([x_1,\ldots ,x_m]\) will denote the convex hull of points \(x_1,\ldots ,x_m\in \mathbb R^d\).
One of the motivations of the Matolcsi–Ruzsa inequality is the observation that to prove (1) for the sumset \(A+A\), the relevant points of the second summand are the vertices of \([A]\).

\(B\subset A\) (Lemma 18).

\((A\cap [B'], B')\) is also \(k\)critical, for any subset \(B'\subset B\) (Lemma 19).

\(B\) is totally stackable (Corollary 21), meaning that all of its triangulations are stacked.
Theorem 3
The \(h\)vector \((h_0,h_1,\dots ,h_d)\in \mathbb N^{d+1}\) (here and in what follows \(\mathbb N=\{0,1,2,\dots \}\)) of a \(d\)dimensional 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 \(h_i\ge 0\) 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 \(h_i=0\) for all \(i\ge 2\), which is equivalent to them having \(Bd\) simplices. Hence, \(B\) needs to be totally stackable.
Corollary 4
The geometric structure of critical pairs is complemented by its arithmetic structure. To express this arithmetic structure we introduce the following concepts. For finite \(B\subset \mathbb R^d\), we write \(\Lambda (B)\) to denote the additive subgroup of \(\mathbb R^d\) generated by \(BB\), and hence by \(B\) if \(0\in B\). We note that \(\Lambda (B)\) is called a lattice if it is of rank \(d\), which will be the typical case.
Definition
The fact that \(A\) is \(B\)stable provides a substantial arithmetic structure to \(A\). For example, suppose that \(A\) is \(B\)stable and let \(l\) be a line intersecting \(A\) and such that \(\Lambda (B)\) contains nonzero vectors parallel to \(l\). Let \(w\) be the shortest such vector (which is unique up to sign). Then \(A\!\cap l\) can be partitioned into arithmetic progressions with common difference \(w\), each of which equals \((x\!+\!\mathbb Zw)\!\cap \! [B]\) for some \(x\!\in \! \mathbb R^d\). 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 \(\ell (e)\!\ge \! \ell (e')\) of \(B\) then the translation of \(A\!\cap e'\) within \(e\) matching one vertex of \(e\) is contained in \(A\!\cap e\).
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
 (i)
\(B=d+1\). That is, \(B\) is the vertex set of a \(d\)simplex.
 (ii)
For \(d\ge 1\), \(B\) consists of the vertices of the simplex \([v_0,\ldots ,v_d]\), and some extra points on the edge \([v_0,v_d]\). The points of \(B\) on this edge are part of an arithmetic progression \(D\) contained in \(A\), and \(A\backslash (B\cup D)\) is the disjoint union of translates of \(D\setminus \{v_0\}\).
 (iii)
For \(d\ge 2\), \([B]\) 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 \(2\le q<d\), \(A\) and \(B\) are the unions of some \(dq\) points and sets \(A', B'\) respectively, where \((A',B')\) is a pair of \(q\)dimensional 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 \(A\setminus A'\) 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 is independent of \(k\). One direction (the fact that \(k\)criticality implies \((k1)\)criticality, if \(k\ge 2\)) 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 \(\Lambda (B)\) 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).
Corollary 6
 (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 \(2\)face of the simplex.
 (iii)
For \(d\ge q\ge 2\), \([A]\) is an iterated pyramid over a \(q\)dimensional simplexprism. There exists a nonzero \(w\in \mathbb R^d\) 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 \(\mathcal T\) of a finite set \(A\) spanning \(\mathbb R^d\) is unimodular if \(\Lambda (A)\) is a lattice with determinant \(\Delta \), and each full dimensional simplex of \(\mathcal T\) has volume \(\Delta /d!\). We note that if \(A\) has a stacked unimodular triangulation then all of its triangulations are unimodular and stacked.
Corollary 7
Let \(k\ge 2\), \(d\ge 2\), and let the finite \(A\) span \(\mathbb R^d\). 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 \(k\)critical for any \(k\ge 1\). Theorem 15 is also the base of the arguments leading to the fundamental properties of \(k\)critical 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 \(0\le b_1<\cdots <b_n\), \(n\ge 2\), we define \(\mathrm{gcd}\{b_1,\ldots ,b_n\}\) to be the largest rational number \(w\) such that \(b_1/w,\ldots ,b_n/w\) are integers. We observe that if \(A,B\subset \mathbb R\) are finite such that \(A\subset [B]= [0,1]\), then \(A\) being stable with respect to \(B\) is equivalent to saying that \(B\subset \mathbb {Q}\), and \(A\) is the union of maximal arithmetic progressions in \([0,1]\) with difference \(w=\mathrm{gcd} (B)\).
Proposition 8
 (i)
The pair \((A,B)\) is \(k\)critical if and only if \(\{ 0,1\}\subset A\) and \(A\) is stable with respect to \(B\).
 (ii)If \(C\subset (0,1)\) is finite, thenwith equality if and only if \(C\) is stable with respect to \(B\).$$\begin{aligned} C+kB\ge (k+1)C, \end{aligned}$$(3)
Proof
We note that if either \(0\not \in A\) or \(1\not \in A\), then a translate of \(A\) is contained in \((0,1)\), a case dealt with in (ii), which shows that \((A,B)\) is not \(k\)critical. Thus let the pair \((A,B)\) be \(k\)critical with \(\{ 0,1\}\subset A\).
3 Some Observations about Triangulations
Throughout this paper, a triangulation of a finite point set \(B\subset \mathbb R^d\) is a geometric simplicial complex with vertex set \(B\) and underlying space \([B]\). A triangulation will be given as a list of \(d\)simplices.
Let \({\mathcal T}=\{S_1,\ldots , S_m\}\) be a triangulation of \(B\). We say that the ordering \(S_1,\ldots , S_m\) of the simplices of \(\mathcal T\) is a shelling if, for every \(i\), the intersection of \(S_i\) with \(S_1\cup \cdots \cup S_{i1}\) is a union of facets of \(S_i\). Equivalently, if \(S_1\cup \cdots \cup S_{i}\) is a topological ball for every \(i\). The index of a simplex \(S_i\) in a shelling is the number of facets of \(S_i\) that are contained in \(S_1\cup \cdots \cup S_{i1}\). That is, the index of \(S_1\) is zero and the index of every other \(S_i\) is an integer between 1 and \(d\). The \(h\)vector of a shelling is the vector \(h=(h_0,\ldots ,h_d)\) with \(h_i\) equal to the number of simplices of index \(i\). We recall without proof some simple facts about shellings and \(h\)vectors (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 \(h\)vector of a shellable triangulation is independent of the choice of shelling. In fact, the \(h\)vector of a (perhaps nonshellable) triangulation can be defined aswhere \((f_{1},\ldots ,f_d)\) is the \(f\)vector of \(\mathcal T\). That is, \(f_i\) is the number of \(i\)simplices in \(\mathcal T\), with the convention that \(f_{1}=1\).$$\begin{aligned} h_k = \sum _{i=0}^k (1)^{ki} {d+1i \atopwithdelims ()ki} f_{i1}, \end{aligned}$$
 (iii)
Every triangulation of \(B\) has \(h_0=1\), \(h_1= Bd1\), and \(\sum h_i=m\), where \(m\) and \(d\) are the number of \(d\)simplices and the dimension of \(\mathcal T\).
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 \(B\subset \mathbb R^d\) be a finite point set and let \(x\in B\) be such that \(B':=B\setminus \{x\}\) is \(d\)dimensional and \(x\not \in [B']\). If \(\mathcal T'\) is a triangulation of \(B'\), we call placing of \(x\) in \(\mathcal T'\) the triangulation \(\mathcal T\) of \(B\) obtained adding to \(\mathcal T'\) the pyramids with apex at \(x\) of all the boundary \((d1)\)simplices of \(\mathcal T'\) that are visible from \(x\). Here, we say that a \((d1)\)simplex \(S\) in the boundary of \(B'\) is visible from \(x\) if its supporting hyperplane \(H\) separates \(x\) from \(B'\setminus H\). Equivalently, if \([x,y]\cap [B'] = \{y\}\) for every point \(y\in S\). It can be shown that if \(\mathcal T'\) is shellable then \(\mathcal T\) is shellable too.
The placing procedure can be used to construct a (shellable) triangulation of \(B\) from scratch, by choosing an initial simplex \(S=[x_1,\ldots ,x_{d+1}]\) with \(\{x_1,\ldots ,x_{d+1}\}\subset B\) and \(S\cap (B\setminus \{x_1,\ldots ,x_{d+1}\}) = \emptyset \), or to extend a given triangulation of a subset \(B'\subset B\) with \([B'] \cap (B\setminus B') = \emptyset \).
Lemma 10
 (i)
The number of \(d\)simplices in \(\mathcal T\) equals \(Bd\).
 (ii)
\(h_i=0\) for all \(i\ge 2\).
 (iii)
The dual graph of \(\mathcal T\) is a tree. The dual graph is the graph having as vertices the \(d\)simplices of \(\mathcal T\) and as edges the adjacent pairs (pairs that share a facet).
 (iv)
Every simplex of dimension at most \(d2\) of \(\mathcal T\) is contained in \(\partial [B]\).
Proof
 (i)
\(S_1\cup \cdots \cup S_{i}\) has at most one vertex more than \(S_1\cup \cdots \cup S_{i1}\). If the total number of vertices equals \(Bd\) we need the number to always increase by one, which implies \((S_1\cup \cdots \cup S_{i1})\) intersects \(S_i\) only in a facet.
 (iii)
If the dual graph is a tree, it has one less edge than vertices. Then, no \(S_i\) has two facets in common with \(S_1\cup \cdots \cup S_{i1}\). It may in principle have a facet plus some lower dimensional face \(\sigma \), but this would imply the dual graph of the link of \(\sigma \) in \(S_1\cup \cdots \cup S_{i}\) 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 \(\sigma \) and is glued to \(S_1\cup \cdots \cup S_{j1}\) along at least two facets, a contradiction.
 (iv)
If every simplex of dimension at most \(d2\) of \(\mathcal T\) is contained in \(\partial [B]\), then every \((d1)\)simplex in \(\mathcal T\) disconnects \(\mathcal T\). Hence the dual graph is a tree and, by the previous argument, \(\mathcal T\) is shellable.\(\square \)
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
 (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 \(d1\) points of \(B\) has \([C]\subset \partial [B]\).
Proof
The implication (ii)\(\Rightarrow \)(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)\(\Rightarrow \)(ii).
Let \(C\subset B\) 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 \(B_F:=B\cap F\) has a triangulation \(\mathcal T_F\) using \(C\) as a simplex. Since \([C]\) goes through the interior of \(F\), the link of \(C\) in \(\mathcal T_F\) is a \((\dim (C)k)\)sphere. In particular, since \(\dim (C)k>0\), its dual graph has cycles. This \(\mathcal T_F\) 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. \(\square \)

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 \(\mathrm{aff }(B')\).
Theorem 12
 (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).
Claim 13
 (i)
\(kC={d+k \atopwithdelims ()k}\).
 (ii)For distinct points \(a,b\in S\) and \(k\ge 1\) we haveunless both \(a,b\in C\).$$\begin{aligned} (a+kC)\cap (b+kC)=\emptyset , \end{aligned}$$
Proof
For (ii), we observe that, for each pair \(x,y\in kC\) of distinct points, the sets \(x+[0,1)^d\) and \(y+[0,1)^d\) are disjoint. Since \(S\backslash \{v_0,\ldots ,v_d\}\subset [0,1)^d\), it follows from \((a+kC)\cap (b+kC)\ne \emptyset \) that either \(a\) or \(b\) is a vertex, say \(a\) is a vertex of \(S\). Then \(x+a\in \mathbb Z^d\), thus \(b\) is a vertex of \(S\) as well. \(\square \)
Corollary 14
Proof
Proof of Theorem 1
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
 (i)
\(C_1\subset A\);
 (ii)
\(A_i+kC_i=(A+kB)\cap (k+1)T_i\) for \(i=1,\ldots ,m\)
We will also use the following consequence of the proof of Theorem 1.
Lemma 16
5 Proof of Sufficiency in Theorem 5
Based on Theorem 15, we show that the pairs \((A,B)\) in Theorem 5 are \(k\)critical for any \(k\ge 1\). 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
Proof
It is sufficient to prove that for any \(a\in A\) and \(b\in B\), there exist \(a'\in A\) and \(b'\in B'\) such that \(a+b=a'+b'\). Since \(B\subset A\), we may assume that \(a,b\not \in B'\). In particular, we may assume that one of the cases (ii)–(iv) holds.
Write \(\{a,b\}=\{\tilde{a},\tilde{b}\}\), where \(a=\tilde{a}\) and \(b=\tilde{b}\) in the cases (ii) and (iv), and vertical edge of \([B]\) containing \(\tilde{b}\) is not longer than the vertical edge containing \(\tilde{a}\) 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 \(\tilde{b}\) belongs to an arithmetic progression \(D\) along an edge \([u,v]\) of \([B]\) containing its two vertices \(u,v\), and \(\tilde{a}\) belongs to \(x+D\backslash \{u\}\subset A\) for some \(x\). The result follows since \(D+D\backslash \{u\}=\{ u,v\}+D\backslash \{u\}\). \(\square \)
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 \(d2\) points to a pair \((A',B')\) described in the case (iv).
Therefore we assume that \([B]\) is an iterated pyramid over a \(q\)dimensional simplexprism \(P\), \(2\le q\le d\), and referring to Lemma 17, also that \(B\) consists of the vertices of \([B]\). Let \(B_0= B\backslash (B\cap P)\). We write \(v_1,\ldots ,v_q,w_1,\ldots ,w_q\) to denote the vertices of \(P\) in a way such that the vectors \(w_iv_i\) are parallel pointing into the same direction for \(i=1,\ldots ,q\). We define \(S_i=[\{v_1,\ldots ,v_i, w_i\ldots w_q\}\cup B_0]\) for \(i=1,\ldots ,q\), and hence \(S_1,\ldots ,S_q\) form a shelling of the corresponding triangulation of \(B\). We write \(A_i\), \(C_i\), \(T_i\) to denote the corresponding sets defined in Theorem 15 for \(i=1,\ldots ,q\).
6 Basic Properties of \(k\)Critical Pairs
The goal of the section is to prove Theorem 23 listing some fundamental properties of \(k\)critical pairs. The first one is a direct consequence of Theorem 15.
Lemma 18
If \((A,B)\) is a \(k\)critical pair, \(k\ge 1\), then \(B\subset A\).
Proof
For any \(x\in B\), we consider a shellable triangulation \(\mathcal T\) with \(x\in C_1\) for the first simplex \(S_1=[C_1]\) of \(\mathcal T\) (this can be achieved, for example, via a placing triangulation). Theorem 15 yields \(C_1\subset A\), thus \(x\in A\). \(\square \)
Based on Theorem 15, we prove that criticality is preserved by taking subsets of \(B\).
Lemma 19
Let \(A, B\) be \(d\)dimensional point sets with \(A \subset [B]\), and let \(k\ge 1\). If \((A,B)\) is \(k\)critical, then \((A\cap [B'],B')\) is also \(k\)critical for every \(B'\subset B\).
Proof
Let \(\widetilde{B}=B\cap [B']\) and \(\widetilde{A}=A\cap [B']\). Since \(B'\subset \widetilde{B}\) and \([B']= [\widetilde{B}]\), it is sufficient to prove that \((\widetilde{A},\widetilde{B})\) is \(k\)critical.
If \(\mathrm{dim}\,[B']=d\), then constructing a placing triangulation first for \(\widetilde{B}\), we obtain some shelling \(S_1,\ldots ,S_m\) of a triangulation of \(B\) such that the union of \(S_1,\ldots ,S_n\) is \([B']\) for some \(n\le m\). Now Theorem 15 (i) and (ii) for the pair \((A,B)\) readily yield the analogous properties for the pair \((\widetilde{A},\widetilde{B})\).
Next we assume that \(\mathrm{dim}[B']=q<d\). We choose \(x_1,\ldots ,x_{dq}\in B\) such that for \(B^*=\{x_1,\ldots ,x_{dq}\}\cup \widetilde{B}\), we have \(\mathrm{dim}[B^*]=d\) and \(B\cap [B^*]=B^*\). Let \(A^*=A\cap [B^*]\). We observe that \(L=\mathrm{aff}\,B'\) is a supporting \(q\)plane to \([B^*]\), with \(\widetilde{A}=A^*\cap L\), \(\widetilde{B}=B^*\cap L\) and \([\widetilde{B}]=[B^*]\cap L\).
Let \(\widetilde{S}_1,\ldots ,\widetilde{S}_n\) be a shelling of some triangulation of \(\widetilde{B}\), and let \(\widetilde{A}_i\), \(\widetilde{C}_i\), \(\widetilde{T}_i\) for \(i=1,\ldots ,n\) be the corresponding sets for Theorem 15. We need to prove that they satisfy Theorem 15. Since \(B\subset A\) according to Lemma 18, Theorem 15 (i) readily follows, and all we have to verify is Theorem 15 (ii).
Under the assumption \(B\subset A\), 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
Proof
Corollary 21
If the pair \((A,B)\) is \(k\)critical with \(\mathrm{dim}[B]=d\), then \(B\) is totally stackable. In particular, \(B\) is contained in the union of the edges of \([B]\).
Proof
We have \(B\subset A\) according to Lemma 18. Theorem 20 yields that every shellable (in particular, every regular) triangulation has \(h_2=0\). According to the characterization of the \(h\)vectors by Stanley [8] (see also Theorem 8.34 in Ziegler [10]), we have \(h_j=0\) for \(j\ge 2\), which, by Lemma 10, implies that \(B\) is stacked. That is, every regular triangulation of \(B\) has \(Bd\) \(d\)simplices. It is a fact (see [4, Theorem 8.5.19]) that then all the triangulations (regular or not) have the same number \(Bd\) of \(d\)simplices. That is, \(B\) is totally stackable. \(\square \)
Next we prove that equality in Theorem 1 is preserved under reducing the value of \(k\).
Lemma 22
If \((A,B)\) is \(k\)critical for \(k\ge 2\) with \(\mathrm{dim}[B]=d\), then it is also \(k'\)critical for every \(k'=1,\ldots ,k1\).
Proof
Let \(S_1,\dots ,S_m\) 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).
We summarize Lemmas 18, 19 and 22 and Corollary 21 as follows.
Theorem 23
 (i)
\(B\subset A\);
 (ii)
\((A\cap [B'],B')\) is also \(k\)critical for every \(B'\subset B\);
 (iii)
\(B\) is totally stackable, thus \(B\) is contained in the union of the edges of \([B]\);
 (iv)
\((A,B)\) is \(1\)critical, and hence \(A+B= (d+1)A  d(d+1)/2\).
From now on we consider \(k\)critical pairs \((A,B)\) for \(k=1\), which will be simply called critical pairs. Theorem 23 (iv) shows that \(k\)critical pairs are critical. We also speak about critical sets in the case of the one dimensional version \(A+B\ge 2A1\) of the Matolcsi–Ruzsa inequality.
7 The Case of a Simplex
In this section we consider the case where \([B]\) is a \(d\)simplex. First we discuss iterated pyramids, a case that will be used later on as well.
Lemma 24
 (i)
\((A\cap [B_0],B_0)\) is a critical pair, and
 (ii)
\((A\cap L)+B_0=(q+1)A\cap L\) for any affine \(q\)plane \(L\) parallel to \(L_0=\mathrm{aff} (B_0)\) intersecting \(A\) and avoiding the vertices of \([B]\).
Proof
We have \(B\subset A\) by Theorem 23 (i), and the pair \((A\cap [B_0],B_0)\) is critical by Theorem 23 (ii). Let \([B]=[x_1,\ldots ,x_{dq},B_0]\), and let \(\widetilde{B}\) be the join of \(\{x_1,\ldots ,x_{dq}\}\) and \(B_0\). In particular, \((A,\widetilde{B})\) is a critical pair, by Lemma 19.
We recall that an edge of \([B]\) is loaded if it contains at least three points of \(B\).
Proposition 25
 (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 \(A\backslash (B\cup D)\) is the disjoint union of translates \(D\setminus \{v\}\); 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 \(T\).
Proof
The facts that \(B\subset A\) and \(B\) is contained in the edges of \([B]\) follow from Theorem 23 (i) and (iii). Let \(B'\) be the vertex set of \([B]\).
We may assume \(B>d+1\), and hence there exists a loaded edge \([u,v]\) of \([B]\).
Therefore we may assume \([v_0,v_1]\) and \([v_0,v_2]\) are two loaded edges of \(B\) with \(v_0=0\), and let \(T=[v_0,v_1,v_2]\) be the \(2\)face containing these two edges. In particular, \((A\cap T, B\cap T)\) is a critical pair by Theorem 23 (ii). It follows by (22) that for \(i=1,2\), \(A\cap T\) contains an arithmetic progression \(D_i\) of length \(m_i\ge 3\) with endpoints \(v_0\) and \(v_i\). According to (22), \(a_1=\frac{m_12}{m_11}\,v_1\) is part of a translate of \(D_2\backslash \{v_2\}\) contained in \(A\cap T\), and hence also of a segment \(\sigma \subset T\) of length at least \(\frac{m_22}{m_21}\,\Vert v_2\Vert \). Since \(\frac{m_i2}{m_i1}\ge \frac{1}{2}\), we deduce that \(m_1=m_2=3\), \(D_1=\{v_0,a_1,v_1\}\) and \(D_2=\{v_0,a_2,v_1\}\) for \(a_2=\frac{1}{2}\,v_2\). It follows by (22) that \(a_0=\frac{1}{2}(v_1+v_2)=a_1+a_2\in A\).
Let \(a\in A\backslash (D_1\cup B')\), and let \(L=a+\mathrm{lin}\{v_1,v_2\}\). It follows by (22) applied to the edge \([v_0,v_1]\) that \(a\in \{p,pa_1\}\) where \(\{p,pa_1\}\subset A\). Now \(p\not \in (D_2\cup B')\), and hence applying (22) to the edge \([v_0,v_2]\), we conclude that either \(p+a_2\in A\), or \(pa_2\in A\). In other words, either \([p,pa_1,p+a_2]\subset [B]\cap L\), or \([p,pa_1,pa_2]\subset [B]\cap L\). Since \([B]\cap L\) is a translate of \(\lambda T\) for \(\lambda \in (0,1]\) and \(\{p,pa_1\}\cap D_1=\emptyset \), we deduce that \(\lambda =1\), and \(\{p,pa_1,pa_2\}=\{a_0,a_2,a_1\}\). Therefore \(A\) consists of the vertices of \([B]\), and the midpoints of \(T\). \(\square \)
8 Critical Pairs \((A,B)\) with \(\mathrm{dim}[B]=2\)
Proposition 26
 (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 \(e_{1}=[v_0,v_1]\) and \(e_{2}=[v_2,v_3]\), and each of \(A\cap e_i\) can be partitioned into maximal arithmetic progressions with common difference \(w\); in addition, if \(\ell (e_2)\le \ell (e_1)\) and \([v_0,v_2]\) is an edge of \([B]\), then \((A\cap e_2)(v_2v_0)= A\cap (e_2(v_2v_0))\).
Proof
Let \(B'=\{v_0,v_1,v_2,v_3\}\) be the vertices of \(B\), where we may assume that \(v_0=o\) and \([0,v_1], [0, v_2]\) are edges of \([B]\). By Theorem 23, we may assume that \(B\subset A\) and that \((A,B')\) is critical.
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 \(P_v\) be the convex hull of the other four vertices of \(P\). It follows again by Theorem 23 that \((A\cap P_v,B\cap P_v)\) is a critical pair, as well, and hence Proposition 26 yields that \(P_v\) is a trapezoid.
Since the sum of the angles of \(P\) is \(3\pi \), there exists a side \(f\) of \(P\) such that the sum of the angles at the two endpoints is at least \(\frac{6\pi }{5}>\pi \). Let \(e\) be the diagonal of \(P\) not meeting \(f\), and let \(v\) be the vertex not in \(e\cup f\). It follows that \(P_v\) is a trapezoid where \(e\) and \(f\) are parallel, and \(\ell (e)>\ell (f)\). We deduce from Proposition 26 that there exists \(x\in A\cap e\) different from the endpoints of \(e\).
Now let \(w\) be an endpoint of \(f\). Since \(e\) is a diagonal of \(P_w\), we have \(x\in A\cap \mathrm{int}P_w\). However Proposition 26 (i) and (ii) applied to the pair \((A\cap P_w,B\cap P_w)\) shows that \(A\cap \mathrm{int}P_w=\emptyset \), which is a contradiction. \(\square \)
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
Proof
Combining Lemmas 24 and 28 yield the following.
Lemma 29
If \(d>q\ge 2\) and \((A,B)\) is a critical pair such that \(B\) spans \(\mathbb R^d\), \(A\subset [B]\), and \([B]=[x_1,\ldots ,x_{dq}, P]\) for a \(q\)dimensional simplexprism \(P\), then \(A=\{x_1,\ldots ,x_{dq}\}\cup (A\cap P)\).
It remains to describe the structure of \(A\) and \(B\) when \([B]\) is a simplexprism.
Proposition 30
 (i)
the vertical edges of \([B]\) are parallel,
 (ii)
\(A\) is contained in the vertical edges of \([B]\),
 (iii)
there exists a vertical vector \(w\ne 0\) such that for each vertical edge \(e\), \(A\cap e\) 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 \(B\cap e\). In addition if \(e\) and \(f\) are vertical edges, and \(e+v\subset f\) in a way such that \(e+v\) and \(f\) share a common endpoint, then \((A\cap e)+v=A\cap (e+v)\).
Proof
We have \(B\subset A\) by Theorem 23 (i). Let \([v_0,\ldots ,v_{d1}]\) and \([w_0,\ldots ,w_{d1}]\) be the facets of \([B]\) such that \([v_i,w_i]\) are the vertical edges for \(i=0,\ldots ,d1\).
We prove (ii) again by contradiction, therefore we suppose that there exists an \(x\in A\) not contained in the vertical edges.
The last property (iii) follows from (30) and Proposition 26. \(\square \)
10 Proof of Necessity in Theorem 5
Let \((A,B)\) be a \(k\)critical pair for some \(k\ge 1\) with \(\mathrm{dim}[B]=d\). In particular, \((A,B)\) a \(1\)critical 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 \(d\ge 2\).
Therefore let \([B]\) be an iterated pyramid over \(P\) with \(\mathrm{dim}P=q\) where \(P\) is polygon or a projective deformation of a simplexprism. We may assume that \(P\) is not a triangle. Since the pair \((A\cap P,B\cap P)\) is \(1\)critical 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 \(q\ge 3\).
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 \(q\ge 3\).
Notes
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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