The maximum number of faces of the Minkowski sum of three convex polytopes

We derive tight expressions for the maximum number of $k$-faces, $0\le k\le d-1$, of the Minkowski sum, $P_1+P_2+P_3$, of three $d$-dimensional convex polytopes $P_1$, $P_2$ and $P_3$, as a function of the number of vertices of the polytopes, for any $d\ge 2$. Expressing the Minkowski sum of the three polytopes as a section of their Cayley polytope $\mathcal{C}$, the problem of counting the number of $k$-faces of $P_1+P_2+P_3$, reduces to counting the number of $(k+2)$-faces of the subset of $\mathcal{C}$ comprising of the faces that contain at least one vertex from each $P_i$. In two dimensions our expressions reduce to known results, while in three dimensions, the tightness of our bounds follows by exploiting known tight bounds for the number of faces of $r$ $d$-polytopes, where $r\ge d$. For $d\ge 4$, the maximum values are attained when $P_1$, $P_2$ and $P_3$ are $d$-polytopes, whose vertex sets are chosen appropriately from three distinct $d$-dimensional moment-like curves.


Introduction
We study the Minkowski sum of three d-dimensional convex polytopes, or, simply d-polytopes, and derive tight upper bounds for the number of its k-faces, for 0 ≤ k ≤ d−1, with respect to the number of vertices of the summands. Given two convex polytopes P 1 and P 2 , their Minkowski sum P 1 + P 2 is the set {p 1 + p 2 | p 1 ∈ P 1 , p 2 ∈ P 2 }. This definition extends to any number of summands and also, to non-convex sets of points. The Minkowski sum of convex polytopes is itself a convex polytope, namely, the convex hull of the Minkowski sum of the vertices of its summands.
Minkowski sums are widespread operations in Computational Geometry and find applications in a wide range of areas such as robot motion planning [Lat91], pattern recognition [TRH00], collision detection [LM04], Computer-Aided Design, and, very recently, Game Theory. They reflect geometrically some algebraic operations, and capture important properties of algebraic objects, such as polynomial systems. This makes them especially useful in Computational Algebra, see e.g., [GS93,Stu96,CLO05].
The geometry of the Minkowski sum can be derived from that of its summands: its normal fan is the common refinement of the normal fans of the summands (see [Zie95] for definitions and details). However, its combinatorial structure is not fully understood, partially due to the fact that most algorithms for computing Minkowski sums have focused on low dimensions (see, e.g., [Fog08] for algorithms computing Minkowski sums in three dimensions). The recent development of algorithms that target high dimensions [Fuk04], has led to a more extensive study of their properties (see, e.g., [Wei07]).
A natural and fundamental question regarding the combinatorial properties of Minkowski sums, concerns their complexity measured as a function of the vertices, or the facets of the summands. A complete answer, in terms of the number of vertices or facets of the summands, does not yet exist although for certain classes of polytopes the question has been resolved (see Section 1). Most of the known results offer tight bounds with respect to the number of vertices of the summands; deriving tight upper bounds with respect to the number of facets seems much harder. Knowing the complexity of Minkowski sums is crucial in developing algorithms for their computation, since it allows to quantify their efficiency.
Preliminaries. Let P be a d-polytope; its dimension is the dimension of its affine span. The faces of P are ∅, P , and the intersections of P with its supporting hyperplanes. The former faces are called improper while the latter faces are called proper. Each face of P is itself a polytope, and a face of dimension k is called a k-face. Faces of P of dimension 0, 1, d − 2 and d − 1 are called vertices, edges, ridges, and facets, respectively.
A d-dimensional polytopal complex, or simply d-complex, C is a finite collection of polytopes in R d such that (i) ∅ ∈ C, (ii) if P ∈ C then all the faces of P are also in C and (iii) the intersection P ∩ Q for two polytopes P and Q in C is a face of both. The dimension dim(C) of C is the largest dimension of a polytope in C. A polytopal complex is called pure if all its maximal (with respect to inclusion) faces have the same dimension. In this case the maximal faces are called the facets of C. A polytopal complex is simplicial if all its faces are simplices. Finally, a polytopal complex C is called a subcomplex of a polytopal complex C if all faces of C are also faces of C. For a polytopal complex C, the star of v in C, denoted star(v, C), is the subcomplex of C consisting of all faces that contain v, and their faces. The link of v, denoted by C/v, is the subcomplex of star(v, C) consisting of all the faces of star(v, C) that do not contain v.
One important class of polytopal complexes arises from polytopes. More precisely, a dpolytope P , together with all its faces and the empty set, form a d-complex, denoted by C(P ). The only maximal face of C(P ), which is clearly the only facet of C(P ), is the polytope P itself. Moreover, all proper faces of P form a pure (d−1)-complex, called the boundary complex C(∂P ), or simply ∂P , of P . The facets of ∂P are just the facets of P .
For a d-polytope P , or its boundary complex ∂P , we can define its f -vector as f (P ) = (f −1 , f 0 , f 1 , . . . , f d−1 ), where f k = f k (P ) denotes the number of k-faces of P and f −1 (P ) := 1 corresponds to the empty face of P . From the f -vector of P we define its h-vector as the vector h(P ) = (h 0 , h 1 , . . . , h d ), where h k = h k (P ) = k i=0 (−1) k−i d−i d−k f i−1 (P ), 0 ≤ k ≤ d. Let C be a pure simplicial polytopal d-complex. A shelling S(C) of C is a linear ordering F 1 , F 2 , . . . , F s of the facets of C such that for all 1 < j ≤ s the intersection, F j ∩ j−1 i=1 F i , of the facet F j with the previous facets is non-empty and pure (d − 1)-dimensional. In other words, for every i < j there exists some < j such that the intersection F i ∩ F j is contained in F ∩ F j , and such that F ∩ F j is a facet of F j .
Every pure polytopal complex that has a shelling is called shellable. In particular, the boundary complex of a polytope is always shellable (cf. [BM71]). Consider a pure shellable simplicial polytopal complex C and let S(C) = {F 1 , . . . , F s } be a shelling order of its facets. The restriction R(F j ) of a facet F j is the set of all vertices v ∈ F j such that F j \ {v} is contained in one of the earlier facets. 1 The main observation here is that when we construct C according to the shelling S(C), the new faces at the j-th step of the shelling are exactly the vertex sets G with R(F j ) ⊆ G ⊆ F j (cf. [Zie95, Section 8.3]). Moreover, notice that R(F 1 ) = ∅ and R(F i ) = R(F j ) for all i = j.
Previous work. The complexity of Minkowski sums depends on the geometry of their summands. Worst-case tight upper bounds offer the best possible alternative when the geometric characteristics of a specific instance of the problem are not accounted for. Gritzman and Sturmfels [GS93] have been the first to derive tight upper bounds for the number of k-faces of P 1 + · · · + P r , namely: where m denotes the number of non-parallel edges of P 1 , . . . , P r . Equality occurs when P i are generic zonotopes, i.e., when each P i is a Minkowski sum of edges, and the generating edges of all polytopes are in general position. Our knowledge of tight upper bounds for f k (P 1 + · · · + P r ) as a function of the number of vertices or facets of the summands is much more limited, while the problem of finding such tight bounds is far from being fully understood and resolved. Given two polygons P 1 , P 2 in two dimensions, with n 1 , n 2 vertices (or edges) respectively, their Minkowski sum can have at most n 1 +n 2 vertices; clearly, this bound holds also for the number of edges of P 1 +P 2 , and generalizes in the obvious way for any number of summands (cf. [dBvKOS00]).
In three or more dimensions, Fukuda and Weibel [FW07] have shown what they call the trivial upper bound : given r d-polytopes P 1 , P 2 , . . . , P r in R d , where d ≥ 3 and r ≥ 2, we have f k (P 1 + P 2 + · · · + P r ) ≤ Φ k+r (n 1 , n 2 , . . . , n r ), where n i is the number of vertices of P i , 1 ≤ i ≤ r, and Φ (n 1 , n 2 , . . . , n r ) = 1≤s i ≤n i s 1 +...+sr= In the same paper, Fukuda and Weibel have shown that the trivial upper bound is tight for: (i) d ≥ 4, 2 ≤ r ≤ d 2 and for all 0 ≤ k ≤ d 2 − r, and (ii) for the number of vertices, f 0 (P 1 + P 2 + · · · + P r ), of P 1 + P 2 + · · · + P r , when d ≥ 3 and 2 ≤ r ≤ d − 1. For r ≥ d, Sanyal [San09] has shown that the trivial bound for f 0 (P 1 + P 2 + · · · + P r ) cannot be attained, since in this case: f 0 (P 1 + P 2 + · · · + P r ) ≤ 1 − 1 Karavelas and Tzanaki [KT11] recently extended the range of of d, r and k for which the trivial upper bound (1) is attained. More precisely, they showed that for any d ≥ 3, 2 ≤ r ≤ d − 1 and for all 0 ≤ k ≤ d+r−1 2 − r, there exist r neighborly d-polytopes P 1 , P 2 , . . . , P r in R d , for which the number of k-faces of their Minkowski sum attains the trivial upper bound. Recall that a d-polytope P is neighborly if any subset of d 2 or less vertices is the vertex set of a face of P . Tight bounds for f 0 (P 1 + P 2 + · · · + P r ), where r ≥ d, have very recently been shown by Weibel [Wei12], namely: where S r j is the family of subsets of {1, 2, . . . , r} of cardinality j, and α = 2(d − 2 d 2 ). Tight bounds for all face numbers, i.e., for all 0 ≤ k ≤ d − 1, expressed as a function of the number of vertices or facets of the summands, are known only for two d-polytopes when d ≥ 3. Fukuda and Weibel [FW07] have shown that, given two 3-polytopes P 1 and P 2 in R 3 , the number of k-faces of P 1 + P 2 , 0 ≤ k ≤ 2, is bounded from above as follows: where n i is the number of vertices of P i , i = 1, 2. These bounds are tight. Weibel [Wei07] has derived analogous tight expressions in terms of the number of facets m i of P i , i = 1, 2: Weibel's expression for f 2 (P 1 + P 2 ) (cf. rel. (3)) has been generalized to the number of facets of the Minkowski sum of any number of 3-polytopes by Fogel, Halperin and Weibel [FHW09]; they have shown that, for r ≥ 2, the following tight bound holds: where m i = f 2 (P i ), 1 ≤ i ≤ r. Finally, Karavelas and Tzanaki [KT12] have shown that for any two d-polytopes P 1 and P 2 in R d , where d ≥ 4, and for all 1 ≤ k ≤ d, we have: where n i = f 0 (P i ), i = 1, 2, and C d (n) stands for the cyclic d-polytope with n vertices. The bounds in (4) have been shown to be tight, and match the corresponding, previously known, bounds for 2-and 3-polytopes (cf. rel. (2)).
Overview. In this work we continue the line of research in [KT12], extending the methods to the case of three d-polytopes in R d . This turns out to be far from trivial. Allowing just one more summand significantly raises the problem's intricacy. On the other hand, the case of three d-polytopes provides a valuable insight towards our ultimate goal, the general case of r d-polytopes, for any d, r ≥ 2. Using the tools and methodology applied in this paper, some of the results obtained here can be generalized to the case d, r ≥ 2 (see Section 7), while others still remain elusive. We state our main result, also presented in Theorem 14. Let P 1 , P 2 and P 3 be three d- Then, for all 1 ≤ k ≤ d, we have: . Moreover, for any d ≥ 2, there exist three d-polytopes in R d for which the bounds above are attained.
To establish the upper bounds we first lift the three d-polytopes in R d+2 using an affine basis of R 2 , and form the convex hull C of the embedded polytopes in R d+2 . C is known as the Cayley polytope of the P i 's (see Section 2). Exploiting the bijection between the set F [3] , consisting of the k-faces of C that contain vertices from each P i , and the (k − 2)-faces of P 1 + P 2 + P 3 , we reduce the derivation of upper bounds for f k−2 (P 1 + P 2 + P 3 ) to deriving upper bounds for The rest of our proof follows the main steps of McMullen's proof of the Upper bound Theorem for polytopes [McM70]. In Section 3 we add auxiliary vertices to appropriate faces of the Cayley polytope C, resulting in a simplicial polytope Q whose face set contains F [3] . We then consider the f -vector f (∂Q) and the h-vector h(∂Q) of ∂Q and derive expressions for their entries via the corresponding vectors for F [3] . Using these expressions, we continue by deriving Dehn-Sommerville-like equations for F [3] . As an intermediate step we define the subcomplex K [3] of C as the closure under subface inclusion of F [3] , and derive expressions for its f -and h-vectors (cf. relations (5) and (12) with R = [3]). This allows us to write the Dehn-Sommerville-like equations for F [3] in the very concise form: In Section 4 we establish a recurrence relation for the elements of h(F [3] ) (see Lemma 7). Our starting point is a well known relation by McMullen (cf. rel. (17)), and the expressions for the h-vector of ∂Q already established in the previous section. The recurrence relation for the elements of h(F [3] ) is then used in Section 5 to prove upper bounds on the elements of h(F [3] ) and h(K [3] ). These upper bounds combined with the Dehn-Sommerville-like equations for F [3] , yield refined upper bounds for the values h k (F [3] ) when k > d+2 2 . We end by establishing our upper bounds on the number of k-faces, 0 ≤ k ≤ d − 1, of the Minkowski sum of three d-polytopes by computing f (F [3] ) from h(F [3] ). At the same time we establish conditions on a subset of the elements of the vectors f (F R ), ∅ ⊂ R ⊆ [3], that are sufficient and necessary in order for the upper bounds in the number of k-faces of P 1 + P 2 + P 3 to be tight for all k (F R stands for the set of faces of C that have at least one vertex from each P i for all i ∈ R).
In Section 6 we describe the constructions that establish the tightness of our upper bounds. For d = 2 and d = 3 we rely on previous results. For d ≥ 4 we define three convex d-polytopes, whose vertices lie on three distinct moment-like d-curves, and show that the sets F R , ∅ ⊂ R ⊆ [3], associated with them satisfy the sufficient and necessary conditions mentioned above. We conclude with Section 7, where we discuss the case of four or more summands and directions for future work.

The Cayley trick
Recall that [3] stands for the set {1, 2, 3}, and denote by S 3 j := {R ⊆ [3] | |R| = j}, the set of all subsets of [3] of cardinality j, for 1 ≤ j ≤ 3. To keep the notation lean, in the rest of this paper we shall denote S 3 j as S j . Consider three d-polytopes P 1 , P 2 and P 3 in R d , and choose the basis e 2,1 = (0, 0), e 2,2 = (1, 0), e 2,3 = (0, 1), as the preferred affine basis of R 2 . The Cayley embedding of the P i 's is defined via the maps µ i (x) = (e 2,i , x), and we denote by C the (d + 2)-polytope we get by taking the convex hull of the sets where V i is the vertex set of P i . This is known as the Cayley polytope of the P i . Similarly, by taking appropriate affine bases we define the Cayley polytope C R of all polytopes P i , i ∈ R, where R ∈ S j , j = 1, 2. These are the Cayley polytopes of all pairs of P i 's and, trivially, the P i 's themselves. Clearly, C R ≡ P i , for R ∈ S 1 . Moreover, C ≡ C [3] .
For any ∅ ⊂ R ⊆ [3], let V R denote the union of the sets V i , i ∈ R. In the sequel we shall identify C R ⊂ R d+|R|−1 , for all R ∈ S j , j = 1, 2, with the affinely isomorphic and combinatorially equivalent polytope conv(V R ) ⊂ C ⊂ R d+2 . This will allow us to study properties of these subsets of C by examining the corresponding Cayley polytopes which lie in lower dimensional spaces.
We shall denote by F R , ∅ ⊂ R ⊆ [3], the set of proper faces of C, with the property that In other words, F R consists of all the faces of C that have at least one vertex from each We call W the d-flat of R d+2 : W = { 1 3 e 2,1 + 1 3 e 2,2 + 1 3 e 2,3 } × R d , and consider the weighted Minkowski sum 1 3 P 1 + 1 3 P 2 + 1 3 P 3 . Note that this is nothing more than P 1 + P 2 + P 3 , scaled down by 1 3 , hence these two sums are combinatorially equivalent. The Cayley trick [HRS00] says that the intersection of W with C is combinatorially equivalent (isomorphic) to the weighted Minkowski sum 1 3 P 1 + 1 3 P 2 + 1 3 P 3 , hence also to the unweighted Minkowski sum P 1 + P 2 + P 3 (see also Fig. 1). Moreover, every face of P 1 + P 2 + P 3 is the intersection of a face of F [3] with W . This implies that: To compute the upper bounds for the number of k-faces of P 1 + P 2 + P 3 , in the rest of the paper we assume that C is "as simplicial as possible", i.e., all faces of C are simplicial except for the trivial faces of C R , for all ∅ ⊂ R ⊆ [3]. Otherwise, we can employ the so called bottom-vertex triangulation [Mat02], where we triangulate every face of C except the trivial faces of C R for all ∅ ⊂ R ⊆ [3]. The resulting complex is polytopal and all of its faces are simplicial, except from the seven trivial faces above. Moreover, it has the same number of vertices as C, while the number of its k-faces is never less than the number of k-faces of C.
Under the "as simplicial as possible" assumption above, the faces in F R are simplicial. We shall denote by K R the closure, under subface inclusion, of F R , i.e., K R contains all the faces in F R and all the faces that are subfaces of faces in F R . It is easy to see that K R does not contain any of the trivial faces of C S , S ⊆ R, and, thus, K R is a pure simplicial (d + |R| − 2)-complex, whose facets are precisely the facets in F R . It is also clear that F R ≡ K R ≡ ∂P R , for R ∈ S 1 . Moreover, K [3] is the boundary complex ∂C of the Cayley polytope C, except for its three facets (i.e., (d + 1)-faces) C R , R ∈ S 2 , and its three ridges (i.e., d-faces) P i , 1 ≤ i ≤ 3. Figure 1: Schematic of the Cayley trick for three polytopes. The three polytopes P 1 , P 2 and P 3 are shown in red, green and blue, respectively. The polytope 1 3 P 1 + 1 3 P 2 + 1 3 P 3 is shown in black.
. By the definition of K R , F is either a k-face of F R , or a k-face of F S for some nonempty subset S of R. Hence where, in order for the above equation to hold for k = −1, we set f −1 (F R ) = (−1) |R|−1 . In what follows we use the convention that f k (F R ) = 0, for any k < −1 or k > d + |R| − 2.

f -vectors, h-vectors and Dehn-Sommerville-like equations
We are going to define auxiliary vertices in R d+2 not contained in V i , i = 1, 2, 3. For every ∅ ⊂ R ⊂ [3] we add a vertex y R in the relative interior of C R and, following [ES74], we consider the complex arising by taking successive stellar subdivisions of ∂C as follows: (i) we form the complex arising from ∂C by taking the stellar subdivisions st(y {i} , C {i} ) for all 1 ≤ i ≤ 3, then (ii) we form the complex arising from the one constructed in the previous step by taking the stellar subdivisions st(y R , C R ) for every R ∈ S 2 , where C R is the complex obtained by taking, for every S ⊂ R, the stellar subdivision of y S over the boundary complex of C S . This complex is polytopal and isomorphic to the boundary complex of a (d + 2)-polytope which we shall denote as Q (see also Fig. 2). The boundary complex ∂Q is a simplicial (d + 1)sphere. The simpliciality of ∂Q will allow us to utilize its Denh-Sommerville equations in order to prove Dehn-Sommerville-like equations for F [3] in the upcoming Lemma 3. We denote by Let us count the k-faces of ∂Q. Suppose that F is a k-face of ∂Q. We distinguish between the following cases depending on the number of auxiliary vertices, y R , that F contains: (i) F does not contain any additional auxiliary vertices. Then, it can be a k-face of any F R , R ∈ S 1 , or it can be a k-face of any of the F R , R ∈ S 2 , or it can be a k-face of F [3] . This gives a total of f k (F [3] ) (ii) F contains one auxiliary vertex. Then, it can consist of a (k − 1)-face of: (a) F R , R ∈ S 1 and vertex y R , (e.g., a (k − 1)-face of ∂P 1 and vertex y {1} ), or (b) F R , R ∈ S 2 and vertex y R , (e.g., a (k − 1)-face of F {1,2} and vertex y {1,2} ), or (c) F S , S ∈ S 1 and vertex y R , where S ⊂ R ∈ S 2 , (e.g., a (k − 1)-face of ∂P 1 and vertex y {1,2} or vertex y {1,3} ), for a total of faces equal to: (iii) F contains two auxiliary vertices. Then, it can consist of a (k − 2)-face of F R , R ∈ S 1 and vertices y R and y S , where S ∈ S 2 such that R ⊂ S, (e.g., a (k − 2)-face of ∂P 1 and vertices y {1} and either y {1,2} or y {1,3} ), for a total of 2 R∈S 1 f k−2 (F R ) faces.
Summing over all previous cases we obtain the following relation, for all 0 ≤ k ≤ d + 1: Relation (6) also holds for k ∈ {−1, 0}, since, by convention, we have set f l (F S ) = 0 for all l < −1 and ∅ ⊂ S ⊆ [3]. Denote by Y a generic subset of faces of C. Y will either be a subcomplex of the boundary complex ∂C of C, or one of the F R 's. Let δ be the dimension of Y. Then we can define the h-vector of Y as Another quantity that will be heavily used in the rest of the paper is that we call the m-order g-vector of Y, the k-th element of which is given by the following recursive formula: Observe that for m = 0 we get the h-vector of Y, while for m = 1 we get what is typically known as the g-vector of Y. Clearly, g (m) (Y) is the m-order backward finite difference of h(Y), which suggests the following lemma (see Section A.1 of Appendix A for the proof): Lemma 1. For any k, m ≥ 0, we have: We next define the summation operator S k (·; D, ν) whose action on Y is as follows: Regarding the action of S k (·; D, ν) on Y, it is easy to verify the following (see Section A.1 of Appendix A for the proof): In the following lemma we relate the h-vectors of F R and K R with each other, and with the h-vector of ∂Q. The last among the relations proved in the following lemma can be thought of as the analogue of the Dehn-Sommerville equations for F [3] and K [3] .
To prove what we named the Dehn-Sommerville-like equations for F [3] (cf. (14)), we replace k by d + 2 − k in (13), to get, for all 0 ≤ k ≤ d + 2: Using the above relation, in conjunction with (13), the Dehn-Sommerville equations for ∂Q become: Using the Dehn-Sommerville equations for F R , R ∈ S 1 , as well as the Dehn-Sommerville-like equations for F R , R ∈ S 2 (cf. [KT12,rel. (3.10)]), we get: Finally, solving in terms of h d+2−k (F [3] ), we arrive at the following: where for the last equality we used relation (12) for R ≡ [3].

Recurrence relations
Recall that we denote by V the vertex set of ∂Q and by V i the (Cayley embedding of the) vertex McMullen [McM70] showed that for any d-dimensional polytope P the following relation holds: Applying relation (17) to the (d + 2)-dimensional polytope Q, we have, for all 0 ≤ k ≤ d + 1: Lemma 4. The h-vectors of the complexes ∂Q/v, v ∈ V i , i = 1, 2, 3, ∂Q/y R , R ∈ S 1 , and ∂Q/y R , R ∈ S 2 are given by the following relations: Proof. We start by proving relation (19). Without loss of generality we assume that v ∈ V 1 ; the cases v ∈ V 2 and v ∈ V 3 are entirely analogous. Let F be a k-face of ∂Q/v. We have the following cases depending on the number of additional points y R , ∅ ⊂ R ⊂ [3], that F contains: Summing over all previous cases we obtain the following relation: We apply the summation operator S k (·; d, 0) to the d-complex ∂Q/v and obtain: which finally gives, for any v ∈ V 1 : To prove (20) consider a k-face of ∂Q/y R , R ∈ S 1 . Such a face is either a k-face of F R , or consists of a (k − 1)-face of F R and point y S for any S ∈ S 2 such that S ⊃ R. Note that there exactly two such points y S . Hence: Applying the summation operator S k (·; d, 0) to the simplicial d-complex ∂Q/y R , R ∈ S 1 , and using relation (23) and Lemma 2, we get, for any R ∈ S 1 : To prove (21) consider a k-face of ∂Q/y R , R ∈ S 2 . This is either a k-face of F S , for any ∅ ⊂ S ⊆ R, or consists of a (k − 1)-face of F S and point y S for any ∅ ⊂ S ⊂ R. Hence, for any R ∈ S 2 , we have: Applying the summation operator S k (·; d, 0) to the d-dimensional complex ∂Q/y R , R ∈ S 2 , and using relation (24), along with Lemma 2, we get, for any R ∈ S 2 : The following two lemmas are essential in the proof of the upcoming recurrence relation in Lemma 7.
Lemma 5. The following relation holds, for all 0 ≤ k ≤ d + 1: (25) Sketch of proof. The complete proof may be found in Section A.2 of Appendix A. Our starting point is relation (18). We first substitute h k (∂Q) and h k+1 (∂Q) on the left-hand side of (18) with their relevant expressions from (13). We then group the terms so that we get a sum of: (i) the left-hand side of (25), (iv) additional terms.
As will be described below, the intuition behind this grouping is to substitute the terms in (ii) and (iii) by sums involving quantities of the form g (m) k (K S /v). These quantities will be grouped with the terms obtained from a similar expansion of the term h k (∂Q/v) appearing in the right-hand side of (18), yielding the right-hand side of (25).
In the proof of [KT12, Lemma 3.2], the sum in item (ii) above is shown to be equal 2 to For (iii) we use (17) combined with the fact that for any R ∈ S 1 , F R ≡ ∂P R . On the right-hand side of (18) we substitute h k (∂Q/v) and h k (∂Q/y R ) using the relations in Lemma 4. Finally, we equate our expansions of the left-and right-hand side of (18) and notice that the terms in (iv) and the expressions for h k (∂Q/y R ) cancel-out.
we appropriately regroup the remaining terms to obtain the desired expression.
Lemma 6. The following relation holds, for all 0 ≤ k ≤ d + 1: Proof. Let us first observe that, by rearranging terms, we can rewrite relation (26) as follows: Clearly, to show that relation (27) holds, it suffices to prove that: In the rest of the proof we shall prove relation (28) for i = 1 and for any v ∈ V 1 . The cases i = 2 and i = 3 are entirely similar. Fix a vertex v ∈ V 1 . Let ∂Q be the polytopal (d + 1)-complex that we get by removing from ∂Q the faces that are incident to y {2,3} (see Fig. 3(left)). It is straightforward to see that: (1) the stars of v in Q and ∂Q coincide (the faces incident to y {2,3} contain vertices from V {2,3} ∪ {y {2} , y {3} } only), and (2) ∂Q is shellable. To verify the latter consider a shelling S(∂Q) of ∂Q that shells the star of y {2,3} in ∂Q last; the shelling order that we get by removing Figure 3: Left: the (d + 1)-complex ∂Q that we get from ∂Q be removing all faces incident to y {2,3} . Right: the (d + 1)-complex Z that we get from the Cayley polytope C [3] of P 1 , P 2 and P 3 , after we: (i) have performed stellar subdivisions using the vertices y {1} , y {2} and y {3} (which yields the (d + 1)-polytope Z), and (ii) have removed the facet Q {2,3} from Z.
from S(∂Q) the facets that are incident to y {2,3} is clearly a shelling order for ∂Q . Let S R , R ∈ {{1, 2}, {1, 3}}, be the star of y R in ∂Q (which actually coincides with the star of y R in ∂Q). Let X denote the set of faces of ∂Q that are either faces in S {1,2} or faces in S {1,3} , and let G denote the set of faces of ∂Q that are either faces in F [3] or faces in F {2,3} . Notice that the sets X and G form a disjoint union of the faces in ∂Q , which implies that: Notice that X is a (d + 1)-complex, whereas G is a set of faces with maximal dimension d + 1. By applying the summation operator S k (·; d + 1, 0) to (29), we immediately get the corresponding h-vector relation: We claim that there exists a specific shelling S(∂Q ) of ∂Q , which actually is an initial segment of a shelling of ∂Q that shells the star of y {2,3} last, with the property that the corresponding shelling order has the facets in X before the facets in G. We will postpone the proof of this claim, and we will assume for now that the claim holds true. Consider the specific shelling of ∂Q just mentioned, and notice that the facets in G are actually the facets in F [3] . The existence of this particular shelling S(∂Q ) also implies that X is shellable, and the shelling S(X ) of X induced by S(∂Q ) coincides with S(∂Q ) as long as it visits the facets of X . As a result of this, and as long as we shell X , we get a contribution of +1 to both h k (∂Q ) and h k (X ) for every restriction of S(∂Q ) of size k. After the shelling S(∂Q ) has left X , a restriction of size k for S(∂Q ) contributes +1 to h k (∂Q ), does not contribute to h k (X ) (X has already been fully constructed), and, thus, by relation (30), contributes +1 to h k (G). In other words, for this particular shelling S(∂Q ) of ∂Q , h k (G) counts the number of restrictions of size k that correspond to the facets of ∂Q that are also facets of G (and, of course, of F [3] ).
The same argumentation can be applied to the links of vertices v ∈ V 1 : ∂Q /v can be seen as the disjoint union of the sets X /v and G/v, while the particular shelling S(∂Q ) of ∂Q that shells X first, induces a particular shelling S(∂Q /v) for ∂Q /v that shells the facets of ∂Q /v in X /v first. From these observations we immediately arrive at the following h-vector relation for ∂Q /v, X /v and G/v: from which we argue, as above, that h k (G/v) counts the number of restrictions of size k for S(∂Q /v) that correspond to the facets of ∂Q /v that are also facets of Let us now consider the dual graph G ∆ (∂Q) of ∂Q, oriented according to the shelling S(∂Q), as well as the dual graph G ∆ (∂Q/v) of ∂Q/v, also oriented according to the shelling S(∂Q/v). We will denote by V ∆ (Y) the subset of vertices of G ∆ (∂Q) that are the duals of the facets in ∂Q that belong to Y, where Y stands for a subset of the set of faces of ∂Q. Since On the other hand, recall that G is the disjoint union of F [3] and F {2,3} . Using expressions (5), in conjunction with the fact that F S ≡ K S for S ∈ S 1 , we have, for all −1 ≤ k ≤ d + 1: By a similar argument, we can arrive that the following expression for f k (G/v): By applying the summation operators S k (·; d + 1, 0) and S k (·; d, 0) to relations (33) and (34), respectively, we get the corresponding h-vector relations: Relation (28) (for i = 1) follows by substituting the expressions for h k (G) and h k (G/v) from (35) in (32). To finish our proof, it remains to establish our claim that there exists a specific shelling S(∂Q ) of ∂Q with the property that the facets of X appear in the shelling before the facets of G. Let us start with some definitions: we denote by Z the (d + 1)-complex we get by performing the stellar subdivisions on C [3] using the vertices y R , R ∈ S 1 (see also Fig. 3(right)), and by Q R , R ∈ S 2 the (d + 1)-complex that we get by performing stellar subdivisions on the non-simplicial proper faces of C R , namely the faces C S , ∅ ⊂ S ⊂ R. Notice that Q R , R ∈ S 2 , is nothing but a facet of Z, while ∂Q R is actually the link of y R in ∂Q. In fact, we can separate the facets of Z in two categories; they are either (1) facets of the form Q R , R ∈ S 2 , which are non-simplicial, or (2) facets in G (or F [3] ), which are simplicial. Moreover, notice that star(y R , Z), R ∈ S 1 , consists of the faces belonging to the two facets Q S , R ⊂ S ⊂ [3] of Z. Since stellar subdivisions produce polytopal complexes [ES74], Z is polytopal and, thus, shellable. In fact, there exists a particular (line) shelling S(Z) of Z in which the facets of star(y {1} , Z) appear first, while Q {2,3} is the last facet in S(Z). More precisely, for this particular shelling of Z, the two facets Q {1,2} and Q {1,3} appear first, followed by the facets in G, which, in turn, are followed by the facet Let us call Z the (d + 1)-complex we get by removing Q {2,3} from Z. The complex Z is shellable (it follows from the fact that S(Z) has Q {2,3} as its last facet), while the particular line shelling S(Z) of Z described above, yields a shelling S(Z ) for Z in which the facets Q {1,2} and Q {1,3} appear first, followed by the facets in G. Notice that if we perform stellar subdivisions on the two non-simplicial facets Q {1,2} and Q {1,3} of Z (using the vertices y {1,2} and y {1,3} ), we arrive at the simplicial (d + 1)-complex ∂Q described earlier. Furthermore, from the particular shelling S(Z ) of Z described above, we may obtain the sought-for shelling for ∂Q that shells X first and G last. To see this, notice that given any shelling order for ∂P i , i = 1, 2, 3, Using inequality (26) in Lemma 6, we arrive at the following recurrence relation for the elements of h(F [3] ); its proof may be found in Section A.2 in Appendix A.

Upper bounds
In this section we establish upper bounds for the number of (k + 2)-faces of F [3] , 0 ≤ k ≤ d − 1, which immediately yield upper bounds for the number of k-faces of P 1 + P 2 + P 3 . Our starting point is the recurrence relation (36). We shall first prove a few lemmas that establish bounds for the g-vector of F R , R ∈ S 2 , and the h-vectors of F [3] and K [3] .
Proof. The bound clearly holds, as equality, for k = 0. For k ≥ 1, from [KT12, Lemma 3.2] we have: Subtracting h k−1 (F R ) from both sides of (38) we get: Using now the upper bounds for h k−1 (F R ), g k−1 (F S ), ∅ ⊂ S ⊂ R, and noting that n R − d − 2 ≥ 2(d + 1) − d − 2 = d > 0, we deduce, for any k ≥ 1: We focus now on the equality claim. Suppose first that , for λ = k − 1, k, which gives: Suppose now that g k (F R ) = ∅⊂S⊆R (−1) |S| n S −d−3+k k . By relation (39), we conclude that h k−1 (F R ) must be equal to its upper bound (cf. [KT12, Lemma 3.3]), since, otherwise, g k (F R ) would not be maximal, which contradicts our assumption on the value of g k (F R ). This gives: Now the fact that h k (F R ) is maximal, implies that h l (F R ) must be equal to its maximal value for all 0 ≤ l < k. To see this suppose that h l (F R ) is not maximal for some l, with 0 ≤ l < k, and among all such l choose the largest one. Then, Lemmas 3.2 and 3.3 in [KT12] imply that h l+1 (F R ) cannot be maximal, which contradicts the maximality of l. Summarizing, we deduce that if g k (F R ) is equal to its upper bound in (37), so is h l (F R ) for all 0 ≤ l ≤ k. By Lemma 3.3 in [KT12], this implies that f l−1 (F R ) = ∅⊂S⊆R (−1) |S| n S l , for all 0 ≤ l ≤ k.
Lemma 9. For all 0 ≤ k ≤ d + 2, we have: Equality holds for some 0 Proof. We are going to prove relation (40) by induction on k. The result clearly holds for k = 0, since Suppose the bound holds for some k ≥ 0. We will show that it holds for k + 1. Using relation (36), Lemma 8, and the fact that, for any k ≥ 0, n [3] − d − 2 + k ≥ 3(d + 1) − d − 2 = 2d + 1 > 0, we have: where we used the fact that: The rest of the proof is concerned with the equality claim. Assume first that In the above relation we used the combinatorial identity (cf. [GKP89, eq. (5.25)]): Since relation (36) holds for all k ≥ 0, we conclude that h l (F [3] ) must be equal to its upper bound in (40), for all 0 ≤ l < k. To see this suppose that (40) is not tight for some l, with 0 ≤ l < k, and among all such l choose the largest one. Then, relation (36) implies that h l+1 (F [3] ) cannot be equal to its upper bound from (40), which contradicts the maximality of l. Hence, if h k (F [3] ) is equal to its upper bound in (40), so is h l (F [3] ) for all 0 ≤ l < k, which gives, for all l with 0 ≤ l ≤ k: where, in order to get from (41) to (42), we used the combinatorial identity (cf. [GKP89, eq. (5.26)]): We are now going to bound the elements of the h-vector of K [3] . More precisely: Lemma 10. For all 0 ≤ k ≤ d + 2, we have: Furthermore, for d ≥ 3 and d odd, we have: We are now ready to state and prove the main theorem of the paper concerning upper bounds on the number of k-faces of the Minkowski sum of three convex d-polytopes.
Theorem 11. Let P 1 , P 2 and P 3 be three d-polytopes in R d , d ≥ 2, with n i ≥ d + 1 vertices, 1 ≤ i ≤ 3. Then, for all 1 ≤ k ≤ d, we have: where δ = d − 2 d 2 , and n S = i∈S n i . Equality holds for all 1 ≤ k ≤ d, if and only if Proof. If suffices to establish upper bounds for f k (F [3] ) for all 0 ≤ k ≤ d + 1. Indeed, writing the f -vector of F [3] in terms of its h-vector, and using relation (14), along with Lemmas 9 and 10 we get: From Lemma 9 we have: whereas from Lemma 10 we get , where: Our upper bounds follow from the fact that f k−1 (P 1 + P 2 In what follows we concentrate on the necessary and sufficient conditions for the upper bounds in (45) to hold as equalities. From the derivation of the upper bounds above (see also relation (47)), it is clear that the bounds are tight if and only if: According to Lemma 9 and Lemma 10, these conditions are, respectively, equivalent to requiring that: (1) f l−1 (F [3] ) = ∅⊂S⊆[3] (−1) 3−|S| n S l , for all 0 ≤ l ≤ d+2 2 , and (2) f l−1 (F R ) = ∅⊂S⊆R (−1) |R|−|S| n S l , for all 0 ≤ l ≤ min{ d+1 2 , d+|R|−

Tightness of upper bounds
In this section we show that the bounds in Theorem 11 are tight. We distinguish between the cases d = 2, d = 3 and d ≥ 4. For d = 2, it is easy to verify that for k = 1, 2, the right-hard side of inequality (45) evaluates to n 1 + n 2 + n 3 , which is known to be tight.
We then proceed to show that F R , R ∈ S 2 , and F Let x i,j , 1 ≤ j ≤ n i , 1 ≤ i ≤ 3, be n [3] positive real numbers, such that x i,j < x i,j+1 , 1 ≤ j ≤ n i − 1, and let τ be a positive real parameter. Let The value of is chosen such that x i,j < x i,j+1 , for all 1 ≤ j < n i , and for all 1 ≤ i ≤ 3. Finally, we set ζ = τ M , where M ≥ d(d + 1). We are going to define three vertex sets V i as follows: Call P i the d-polytope we get as the convex hull of the vertices in V i , and let V i be the image of V i via the Cayley embedding. As in Section 2, call C the Cayley polytope of the P i 's in R d+2 , and F R , ∅ ⊂ R ⊆ [3], the set of faces of C with at least one vertex from each V i , i ∈ R. Note that, by construction, P i is a d 2 -neighborly polytope in R d with n i vertices, which immediately implies that conditions (46) hold for R ∈ S 1 and for all 0 ≤ l ≤ d 2 . Hence, it suffices to show that: which we will succeed by choosing a sufficiently small value for τ . To prove that the constructed polytopes have the desired properties (see Lemmas 12 and 13, bellow), we adopt the key idea used in the proofs of [Zie95, Theorem 0.7 & Corollary 0.8] on basic properties of cyclic d-polytopes, and adapt this idea to our setting, where we view the faces the Minkowski sum of the polytopes P i , i ∈ R, via the face set F R of their Cayley polytope, where 2 ≤ |R| ≤ 3.
We start off with subsets R of size two. To show that f k−1 (F R ) is according to relation (54), recall (cf. Section 2) that the polytope C contains the Cayley polytope C R of the polytopes in R as a d-subcomplex embedded in R d+2 . Thus, in order to prove relation (54) for F R , we may consider C R and F R independently of C, i.e., we can disassociate the polytopes P i , i ∈ R, from the Cayley polytope C. In other words, we think of the polytopes P i , i ∈ R, as d-polytopes in R d , while their Cayley polytope C R is seen as a (d + 1)-polytope in R d+1 . We exploit this observation in order to prove the following lemma.
Lemma 12. There exists a sufficiently small positive valueτ R for τ such that, for all τ ∈ (0,τ R ), Proof. Without loss of generality let R = {1, 3}. The rest of the cases are analogous. The condition in the statement of the lemma is equivalent to the requirement that C {1,3} is a (V 1 , d+1 2 )bineighborly polytope (see [KT12] for definitions and details), which in turn is equivalent to the requirement that f d+1 We shall prove that condition (55) holds true for the Cayley polytope C {1,3} of the polytopes P 1 , P 3 , and for sufficiently small values of τ , as described in the statement of the lemma.
Define δ := d + 1 − 2 d+1 2 . Let X be a positive real number such that X > x 3,n 3 , and let 4 T = Xτ ν 3 . Choose a set U of k m = 0 vertices γ m (t m,j m,1 ), γ m (t m,j m,2 ), . . . , γ m (t m,j m,km ) from the set V m , such that j m,1 < j m,2 < . . . < j m,km , for m ∈ {1, 3}, and k 1 + k 3 = d+1 2 . Let U = {β m (t m,j m,1 ), β m (t m,j m,2 ), . . . , β m (t m,j m,km ) | m ∈ {1, 3}}, be the Cayley embedding of U in R d+1 (using the affine basis e 1,1 , e 1,2 ). For a vector x = (x 1 , x 2 , . . . , x d+1 ) ∈ R d+1 , we define the (d + 2) × (d + 2) determinant H U (x) as follows: Notice that for d odd the last column , t s,λ = x s,λ τ νs , where 1 ≤ λ ≤ n s , s is either 1 or 3, and λ / ∈ {j s,1 , j s,2 , . . . , j s,ks }. We perform the following determinant transformations on H U (v): initially we subtract its second row from its first, and then we shift its first column to the right via an even number of column swaps. More precisely, we need to shift the first column of H U (v) to the right so that the values t s,λ , t s,j s,1 , t s,j s,1 , t s,j s,2 , t s,j s,2 , . . . , t s,j s,ks , t s,j s,ks appear consecutively in the columns of H U (v) and in increasing order. To do that we always need an even number of column swaps, due to the way we have chosen .
Consider the case where s = 1 and suppose that all necessary operations on H U (v) have been performed. Then H U (v) is in the form of the determinant D n,m (τ ; I, J, µ) of Lemma 16 (multiplied by τ M ), with n ← 2k 1 + 1, m ← 2k 3 , l ← d + 2, µ ← (0, 0, 1, 2, . . . , d), α ← ν 1 , β ← ν 3 , I ← 3, and J ← 5. Note that the requirement for M in Lemma 16 is satisfied by our choice of M . According to Lemma 16, H U (v) has the following asymptotic expansion in terms of τ : where C is a positive constant independent of τ . The asymptotic expansion in (56) implies that there exists a positive valueτ v,U for τ such that for all τ ∈ (0,τ v,U ), H U (v) > 0. The case s = 3 is completely analogous. Since the number of the subsets U is finite, while for each such subset U we need to consider a finite number of vertices in V {1,3} \ U, it suffices to consider a positive valueτ {1,3} for τ that is small enough, so that all possible determinants H U (v) are strictly positive for any τ ∈ (0,τ {1,3} ). For τ ∈ (0,τ {1,3} ), our analysis above immediately implies that for each set U the equation H U (x) = 0, x ∈ R d+1 , is the equation of a supporting hyperplane of C R passing through the vertices of U, and those only. In other words, every set U, where |U| = d+1 2 , |U ∩ V 1 | = k 1 = 0, and |U ∩ V 3 | = k 3 = 0, defines a ( d+1 2 − 1)-face of C R . Taking into account that the number of such subsets U is Hence, condition (55) is satisfied for all τ ∈ (0,τ {1,3} ).
We now consider the case R = [3]. In this case we can show that: Lemma 13. There exists a sufficiently small positive valueτ [3] for τ such that, for all τ ∈ (0,τ [3] ), Proof. Define δ := d + 2 − 2k and let T be a positive real number such that T > t 3,n 3 (= x 3,n 3 ). Choose a set U of k i = 0 vertices from V i , 1 ≤ i ≤ 3, such that k 1 + k 2 + k 3 = k, and denote by U the Cayley embedding of U in R d+2 (using the affine basis e 2,i , be their corresponding vertices in U, where j i,1 < j i,2 < . . . < j i,k i for all 1 ≤ i ≤ 3. Let x = (x 1 , x 2 , . . . , x d+2 ) and define the (d + 3) × (d + 3) determinant H U (x) as follows: x β 1 (t 1,j1,1 ) · · · β 1 (t ǫ 1,j 1,k 1 ) β 2 (t 2,j2,1 ) · · · β 2 (t ǫ We can alternatively describe H U (x) as follows: (i) The first column of H U (x) is 1 x . (ii) For i ranging from 1 to 3, and for λ ranging from 1 to k i , the next k i pairs of columns of H U (x) are 1 β i (t i,j i,λ ) and (iii) For λ ranging from 1 to δ, the last δ columns of H U (x) are 1 β 3 (λT ) . Notice that if k = d+2 2 and d is even, this category of columns of H U (x) does not exist.
The equation H U (x) = 0 is the equation of a hyperplane in R d+2 that passes through the points in U. Recall that V [3] = V 1 ∪ V 2 ∪ V 3 . We are going to show that, for any choice of U, and for all vertices v in V [3] \ U, we have H U (v) > 0 for sufficiently small τ .
Suppose we have some vertex v ∈ V [3] \ U. Then, v = β s (t s,λ ), t s,λ = x s,λ τ νs , for some 1 ≤ λ ≤ n s and 1 ≤ s ≤ 3, such that λ / ∈ {j s,1 , j s,2 , . . . , j s,ks }. Then we can transform H U (v) in the form of the determinant E n,m,k (τ ; µ) of Lemma 17, by subtracting the second and third row of H U (v) from its first row and shifting the first column of H U (v) to the right via an even number of column swaps. More precisely, we need to shift the first column of H U (v) to the right so that the values t s,λ , t s,j s,1 , t s,j s,1 , t s,j s,2 , t s,j s,2 , . . . , t s,j s,ks , t s,j s,ks , appear consecutively in the columns of H U (v) and in increasing order. To do that we always need an even number of column swaps, due to the way we have chosen . Now, suppose that v ∈ V 1 . Then H U (v) is in the form of the determinant E n,m,k (τ ; µ) of Lemma 17, where n ← 2k 1 + 1, m ← 2k 2 , k ← 2k 3 + δ, l ← d + 3, and µ ← (0, 0, 0, 1, 2, . . . , d).
Obviously, M ≥ 2|µ| = d(d + 1). Applying now Lemma 17, we deduce that H U (v) can be written as: where C is a positive constant independent of τ . The asymptotic estimate above implies that H U (v) > 0, for sufficiently small τ . The remaining cases, i.e., the cases v ∈ V 2 and v ∈ V 3 , are completely analogous and we omit them. We thus conclude that, for any specific choice of U , and for any specific vertex v ∈ V [3] \ U, there exists some τ v,U > 0 (cf. Lemma 17) that depends on v and U, such that for 2 , the number of the sets U of size k containing at least one vertex from each V i , 1 ≤ i ≤ 3, is For each such subset U we need to consider the (n 1 + n 2 + n 3 − k) vertices in V [3] \ U, therefore it suffices to consider a positive valueτ [3] for τ that is small enough, so that all d+1 2 k=2 (n 1 + n 2 + n 3 − k) possible determinants H U (v) are strictly positive. For τ ←τ [3] , our analysis above immediately implies that for each set U the equation H U (x) = 0, x ∈ R d+2 , is the equation of a supporting hyperplane for C passing through the vertices of U, and those only. In other words, every set U, of k vertices, for 3 ≤ k ≤ d+2 2 , with at least one vertex from each V i , 1 ≤ i ≤ 3, defines a (k − 1)-face of C, which means that Relation (54) now immediately follows from Lemmas 12 and 13. First choose a value τ for τ , smaller thatτ R , for all 2 ≤ |R| ≤ 3. Then for this value of τ , the results of both Lemma 12 and Lemma 13 hold true. Moreover, since P 1 , P 2 and P 3 are d 2 -neighborly for any τ > 0, and since f −1 (F R ) = (−1) |R|−1 , for all ∅ ⊂ R ⊆ [3], while f k−1 (F R ) = 0, for all 1 ≤ k ≤ |R|, we conclude that, for τ ≡ τ , relations (54) hold.
Based on the analysis above, as well as the analysis in Section 6.1, we conclude that the upper bounds stated in Theorem 11 are actually tight for any d ≥ 2. We can, thus, restate Theorem 11 in its complete and definitive form: Theorem 14. Let P 1 , P 2 and P 3 be three d-polytopes in R d , d ≥ 2, with n i ≥ d + 1 vertices, 1 ≤ i ≤ 3. Then, for all 1 ≤ k ≤ d, we have: where δ = d − 2 d 2 , and n S = i∈S n i . Moreover, for any d ≥ 2, there exist three d-polytopes in R d for which the bounds above are attained for all 1 ≤ k ≤ d.

Summary and open problems
In this paper we have computed the maximum number of k-faces, f k (P 1 +P 2 +P 3 ), 0 ≤ k ≤ d−1, of the Minkowski sum of three d-polytopes P 1 , P 2 and P 3 in R d as a function of the number of their vertices n 1 , n 2 and n 3 . When d = 2 our expressions reduce to known tight bounds, while for d = 3 we show the tightness of our upper bounds by exploiting results from [FW07] and [Wei12]. In four or more dimensions we present a novel construction that achieves the upper bounds: we consider the d-dimensional moment-like curves γ 1 (t) = (t, ζt 2 , ζt 3 , t 4 , . . . , t d ), γ 2 (t) = (ζt, t 2 , ζt 3 , t 4 , . . . , t d ), and γ 3 (t) = (ζt, ζt 2 , t 3 , t 4 , . . . , t d ), and we show that our maximal values are attained when P i is the d-polytope with vertex set with 0 < x i,1 < x i,2 < · · · < x i,n i and ζ = (τ ) M . The parameter value τ is a sufficiently small positive number, while M is chosen sufficiently large. Our ultimate goal is to extend our results for the Minkowski sum of r d-polytopes in R d , for r ≥ 4 and d ≥ 3. Towards this direction, we can extend our methodology and tools so as to prove relations for r polytopes that generalize certain relations that hold true for two or three polytopes. For example, relation (12) in Lemma 3 generalizes to: while the Dehn-Sommerville-like equations in the same lemma (cf. rel. (14)), generalize to: where E m,k , m ≥ k + 1 > 0, are the Eulerian numbers [GKP89,A00]: A recurrence relation similar to (36) in Lemma 7 is not as straightforward to obtain. However, we conjecture that the following recurrence relation holds for all 0 ≤ k ≤ d + r − 2: The bounds presented in this paper refer to polytopes of the same dimension. We would like to derive similar bounds for two or more polytopes when the dimensions of these polytopes differ, as well as in the special case of simple polytopes. Finally, a similar problem is to express the number of k-faces of the Minkowski sum of r d-polytopes in terms of the number of facets of these polytopes. Results in this direction are known for d = 2 and d = 3 only. We would like to derive such expressions for any d ≥ 4 and any number, r, of summands. Proof of Lemma 1. The result clearly holds for m = 0, since: Suppose the relation holds for some m ≥ 0. We will show it holds for m + 1. Indeed: Proof of Lemma 2. By replacing h k−ν−j (Y) from its defining equation, we get: where: • in order to go from (59) to (60), we used that δ+1−i δ+1−k+ν+j = 0 for i > δ + 1, • in order to go from (61) to (62), we used the combinatorial identity: • in order to go from (62) to (63), we used that D+1−ν−i and, finally, • in order to go from (64) to (65), we used that f j−ν−1 (Y) = 0 for j < ν (i.e., for j − ν − 1 < −1), and that D+1−j D+1−k = 0 for j > k.

A.2 Omitted & full proofs of Section 4
Proof of Lemma 5. Using relation (13), and after rearranging the terms, the left hand side of relation (18) becomes: (66) We are going to analyze each term in the expression above separately. For any R ∈ S 2 : (i) the relation at the top of page 18 in [KT12, Lemma 3.2], (ii) relations (12), with R ∈ S 2 , and (iii) relation (3.9) in [KT12], give: Hence term T 2 can be rewritten as: Applying relation (17) to the (d − 1)-complex F R , R ∈ S 1 , and using the identity F R ≡ K R (≡ ∂P R ), we derive the following expressions: which, in turn yield the following expansions for T 3 and T 4 : On the other hand, utilizing the expressions in Lemma 4, we arrive at the following expansion for the right-hand side of (18): the expression in (70) can be rewritten in the following more convenient form: Solving relation (18) in terms of the term T 1 , we get: T 1 = T 12 + T 13 + T 14 + T 15 − (T 2 + T 3 + T 4 ) = T 12 + T 13 + T 14 + T 15 − [(T 5 + T 6 − T 7 ) + (T 8 + T 9 ) + (T 10 + T 11 )] = T 12 + (T 13 − T 6 ) + (T 14 + T 7 − T 8 − T 10 ) + (T 15 − T 5 − T 9 − T 11 ) = T 12 + (T 13 − T 6 ) + (T 14 + T 7 − T 8 − T 10 ), where we used the fact that the terms T 5 , T 9 and T 11 cancel-out with the term T 15 . Observe now that: Hence, Proof of Lemma 7. By Lemma 6, relation (25) yields: By relation (12) with R ≡ [3], we can write h k (K [3] ) as: whereas from relation (12) for all R ∈ S 2 we easily get: Since K R ≡ F R , for any R ∈ S 1 , we can employ relations (72) and (73) to rewrite the right hand side of (71) as follows: Using the identity: we see that the last term (term T ) in the relation above vanishes: Hence, relation (71) simplifies to: from which we obtain the relation in the statement of the lemma.
In what follows we recall some facts concerning generalized Vandermonde determinants that will be in use to us later. Let n ≥ 2, x = (x 1 , . . . , x n ) and µ = (µ 1 , µ 2 , . . . , µ n ), where we require that 0 ≤ µ 1 < µ 2 < . . . < µ n . The generalized Vandermonde determinant, denoted by GVD(x; µ), is the n × n determinant whose i-th row is the vector x with all its entries raised to µ i . While there is no general formula for the generalized Vandermonde determinant, it is a well-known fact that, if the elements of x are in strictly increasing order, then GVD(x; µ) > 0 (for example, see [Gan05] for a proof of this fact).
In the remainder of this section we consider two determinants that are parameterized by a positive parameter τ , and we study their asymptotic behavior with respect to τ . These determinants are generalizations of the determinants that arise in the proofs of Lemmas 12 and 13 in Section 6, and are directly associated with the equations of some appropriately defined supporting hyperplanes for the faces of F R where R ∈ S 2 or R ≡ [3] (recall that F R stands for the set of faces of the Cayley polytope of |R| polytopes P i , i ∈ R, with the property that each face in F R has at least one vertex from each polytope P i ). The two determinants that we study are generalized-Vandermonde-like determinants that are polynomial functions of τ , and correspond, respectively, to the two cases R ∈ S 2 and R ≡ [3] mentioned above. Since in Section 6 we are interested in small values of τ , our asymptotic analysis in the two lemmas below is targeted towards revealing the term of τ of minimal exponent.
We start-off with the generalized version of the determinant that arises in the upper bound tightness construction in Section 6 when R ∈ S 2 .

(79)
The above sum consists of n+m n terms. Among these terms: (i) all those for which r contains the second row vanish (in this case the corresponding row of S(∆ n,m (τ ); r, c) consists of zeros), and (ii) all those for which r does not contain the first row vanish (in this case at least two rows ofS(∆ n,m (τ ); r, c) consist of zeros).
The remaining terms of the expansion are the n+m−2 n−1 terms for which r contains 1 but not 2, i.e., r = (1, r 2 , r 3 , . . . , r n ), with 3 ≤ r 2 < r 3 < . . . < r n ≤ n + m. For any given r, we denote byr the vector of the m, among the n + m, row indices for ∆ n,m (τ ) that do not belong to r (recall that 2 always belongs tor). Notice that the elements of the k-th row of ∆ n,m (τ ) have exponent µ k . Denoting by µ r the vector the i-th element of which is µ r i , we have that: (i) det(S(∆ n,m (τ ); r, c)) is the n × n generalized Vandermonde determinant GVD(τ α x; µ r ), multiplied by τ M if J ∈ r.
We can, thus, simplify the expansion in (79) to get:
In the remainder of the proof we seek to find the unique term in the expansion (80) that corresponds to the minimum order of τ , or, equivalently, the minimum exponent for τ . Since α > β ≥ 0, for any r, with I ∈ r and J ∈ r, the exponent of τ is: where we used the fact that: This implies that the terms in (80) that correspond to the row vectors r that contain J cannot be the terms of minimal order of τ , since for these terms the exponent of τ is at least α|µ r | + β|µr| + M > β|µ r | + β|µr| + M = β|µ| + M ≥ M.
For any given r, we denote byr the vector of the m + k row indices for E n,m,k (τ ) that do not belong to r. Moreover, µ r is the vector the i-th element of which is µ r i . As in the proof of Lemma 16, we seek to find the unique minimum term in the expansion (82) that corresponds to the minimum order of τ , or, equivalently, the minimum exponent for τ .
Since (n − 1)(n + 6) is even for any n, the term in the expansion of E n,m,k (τ ) corresponding to the minimum exponent for τ becomes C ρ τ 2|µρ|+|µρ | GVD(x; µ ρ ). The claim in the statement of the lemma immediately follows from the positivity of C ρ and GVD(x; µ ρ ), and by observing that 2|µ ρ | + |µρ | equals ξ.