f-Vectors of Minkowski Additions of Convex Polytopes

Proceedings of the 29th annual symposium on Symposuim on computational geometry - SoCG '13, 2013

We derive tight expressions for the maximum number of k-faces, 0 ≤ k ≤ 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 ≥ 2. Expressing the Minkowski sum of the three polytopes as a section of their Cayley polytope 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 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 ≥ d. For d ≥ 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.

Discrete & Computational Geometry, 2009

We present a tight bound on the exact maximum complexity of Minkowski sums of polytopes in ℝ3. In particular, we prove that the maximum number of facets of the Minkowski sum of k polytopes with m 1,m 2,…,m k facets, respectively, is bounded from above by $\sum_{1\leq i<j\leq k}(2m_{i}-5)(2m_{j}-5)+\sum_{1\leq i\leq k}m_{i}+\binom{k}{2}$ . Given k positive integers m 1,m 2,…,m k , we describe how to construct k polytopes with corresponding number of facets, such that the number of facets of their Minkowski sum is exactly $\sum_{1\leq i<j\leq k}(2m_{i}-5)(2m_{j}-5)+\sum_{1\leq i\leq k}m_{i}+\binom{k}{2}$ . When k=2, for example, the expression above reduces to 4m 1m 2−9m 1−9m 2+26.

2011

Consider a set of r convex d-polytopes P 1 , P 2 , . . . , P r , where d ≥ 3 and r ≥ 2, and let n i be the number of vertices of P i , 1 ≤ i ≤ r. It has been shown by Fukuda and Weibel [4] that the number of k-faces of the Minkowski sum, P 1 + P 2 + • • • + P r , is bounded from above by Φ k+r (n 1 , n 2 , . . . , n r ), where Φ ℓ (n 1 , n 2 , . . . , n r ) = 1≤si≤ni s1+...+sr =ℓ r i=1 ni si , ℓ ≥ r. Fukuda and Weibel [4] have also shown that the upper bound mentioned above is tight for d ≥ 4, 2 ≤ r ≤ ⌊ d 2 ⌋, and for all 0 ≤ k ≤ ⌊ d 2 ⌋r. In this paper we construct a set of r neighborly d-polytopes P 1 , P 2 , . . . , P r , where d ≥ 3 and 2 ≤ r ≤ d -1, for which the upper bound of Fukuda and Weibel is attained for all 0 ≤ k ≤ ⌊ d+r-1 2 ⌋r. A direct consequence of our result is a tight asymptotic bound on the complexity of the Minkowski sum P 1 + P 2 + • • • + P r , for any fixed dimension d and any 2 ≤ r ≤ d -1, when the number of vertices of the polytopes is (asymptotically) the same. Our approach is based on what is known as the Cayley trick for Minkowski sums: the Minkowski sum, P 1 + P 2 + . . . + P r , is the intersection of the Cayley polytope P, in R r-1 × R d , of the d-polytopes P 1 , P 2 , . . . , P r , with an appropriately defined d-flat W of R r-1 ×R d . To prove our bounds, we construct the d-polytopes P 1 , P 2 , . . . , P r , where d ≥ 3 and 2 ≤ r ≤ d-1, in such a way so that the number of (k-1)-faces of P, that intersect the d-flat W , is equal to Φ k (n 1 , n 2 , . . . , n r ), for all r ≤ k ≤ ⌊ d+r-1 2 ⌋. The tightness of our bounds then follows from the Cayley trick: the (k + r -1)-faces of the intersection of P with W are in one-to-one correspondence with the k-faces of P 1 + P 2 + • • • + P r , which implies that f k (P 1 + P 2 + • • • + P r ) = Φ k+r (n 1 , n 2 , . . . , n r ), for all 0 ≤ k ≤ ⌊ d+r-1 2 ⌋r.

Canadian Conference on Computational Geometry, 2005

We consider the problem of listing faces of the Minkowski sum of several V-polytopes in R d . An algorithm for listing all faces of dimension up to j is presented, for any given 0 ≤ j ≤ d − 1. It runs in time polynomial in the sizes of input and output.

2007

We present a tight bound on the exact maximum complexity of Minkowski sums of convex polyhedra in R 3 . In particular, we prove that the maximum number of facets of the Minkowski sum of two convex polyhedra with m and n facets respectively is bounded from above by f (m, n) = 4mn−9m−9n+26. Given two positive integers m and n, we describe how to construct two convex polyhedra with m and n facets respectively, such that the number of facets of their Minkowski sum is exactly f (m, n). We generalize the construction to yield a lower bound on the maximum complexity of Minkowski sums of many convex polyhedra in R 3 . That is, given k positive integers m1, m2, . . . , m k , we describe how to construct k convex polyhedra with corresponding number of facets, such that the number of facets of their Minkowski sum is P 1≤i<j≤k (2mi − 5)(2mj − 5) +`k 2´+ P 1≤i≤k mi. We also provide a conservative upper bound for the general case. Snapshots of several pre-constructed convex polyhedra, the Minkowski sum of which is maximal, are available at http://www.cs.tau.ac.il/~efif/Mink. The polyhedra models and an interactive program that computes their Minkowski sums and visualizes them can be downloaded as well.

Journal of Symbolic Computation, 2004

By replacing line segments with convex V-polytopes, we obtain a natural generalization of the zonotope construction problem: the construction of the Minkowski addition of k polytopes. Gritzmann and Sturmfels studied this general problem in various aspects and presented polynomial ...

Computing Research Repository, 2009

We end with algorithmic considerations, and we show how our tight bounds for the parallel polytope convex hull problem, yield tight bounds on the combinatorial complexity of the Minkowski sum of two convex polytopes in $\mathbb{E}^d$.

Journal of Combinatorial Theory, Series B, 1971

Monatshefte f�r Mathematik, 1990

We call a convex subset N of a convex d-polytope P c E d a k-nucleus of P if N meets every k-face of P, where 0 < k < d. We note that P has disjoint k-nuclei if and only if there exists a hyperplane in E d which bisects the (relative) interior of every k-face of P, and that this is possible only if/~-/~< k ~< d-1.

One classical result of Freimann gives the optimal lower bound for the cardinality of A + A if A is a d-dimensional finite set in 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.

Computer-aided Design, 2007

We present an exact implementation of an efficient algorithm that computes Minkowski sums of convex polyhedra in R 3 . Our implementation is complete in the sense that it does not assume general position. Namely, it can handle degenerate input, and it produces exact results. We also present applications of the Minkowski-sum computation to answer collision and proximity queries about the relative placement of two convex polyhedra in R 3 . The algorithms use a dual representation of convex polyhedra, and their implementation is mainly based on the Arrangement package of Cgal, the Computational Geometry Algorithm Library. We compare our Minkowski-sum construction with the only three other methods that produce exact results we are aware of. One is a simple approach that computes the convex hull of the pairwise sums of vertices of two convex polyhedra. The second is based on Nef polyhedra embedded on the sphere, and the third is an output sensitive approach based on linear programming. Our method is significantly faster. The results of experimentation with a broad family of convex polyhedra are reported. The relevant programs, source code, data sets, and documentation are available at http://www.cs.tau.ac.il/~efif/CD, and a short movie [16] that describes some of the concepts portrayed in this paper can be downloaded from

Mathematika, 1970

In this paper we give a proof of the long-standing Upper-bound Conjecture for convex polytopes, which states that, for 1 ≤ j…

Discrete Mathematics, 1993

Dedicated to Professor B. Griinbaum on the occasion of his 60th birthday. Abstract Jendrol', S., On face vectors and vertex vectors of convex polyhedra, Discrete Mathematics 118 (1993) 1199144. Denote by p,(M) or v,(M) the number of k-gonal faces, or k-valent vertices, respectively, of the convex polyhedron M. A pair of sequences (p,(M) 1 k > 3) and (u,(M) 1 k > 3) associated in a natural way with a polyhedron M is called the face vector and the vertex vector of M, respectively. Let p=(pL 13 <k f6) and v=(v~ 1 k>4) be a pair of sequences of nonnegative integers satisfying xkss(6-k)p,+2x,,,(3-k)uk=

Journal of the London Mathematical Society, 1971

In this paper we study enumeration problems for polytopes arising from combinatorial optimization problems. While these polytopes turn out to be quickly intractable for enumeration algorithms designed for general polytopes, algorithms using their large symmetry groups can exhibit strong performances. Specifically we consider the metric polytope mn on n nodes and prove that for n ≥ 9 the faces of codimension 3 of mn are partitioned into 15 orbits of its symmetry group. For n ≤ 8, we describe additional upper layers of the face lattice of mn. In particular, using the list of orbits of high dimensional faces, we prove that the description of m8 given in [9] is complete with 1 550 825 000 vertices and that the Laurent-Poljak conjecture [15] holds for n ≤ 8. Computational issues for the orbitwise face and vertex enumeration algorithms are also discussed.