On the Exact Maximum Complexity of Minkowski Sums of Polytopes

Fogel, Efi; Halperin, Dan; Weibel, Christophe

doi:10.1007/s00454-009-9159-1

Cited by 21 publications

(29 citation statements)

References 15 publications

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“…Conversely, each face F of a d-dimensional polytope P can be associated with a region of the sphere S d−1 , called the normal region, which is the set of unit vectors outwardly normal to some support hyperplane of P , whose intersection with P is We call a subset of the sphere S d−1 spherically convex if for any two points in the subset, any shortest arc of great circle between the two points is inside the subset. 1 If the polytope P is full-dimensional, the normal regions of faces of P are all disjoint, relatively open and spherically convex. They determine a subdivision of S d−1 into a spherical cell complex, which we call the Gaussian map of the polytope:…”

Section: Minkowski Sumsmentioning

confidence: 99%

“…Formally, for any points p and q of S d−1 that are not in U , we say that p is to the west of q if θ 1 (p) ∈ [θ 1 (q), θ 1 (q) + π] and θ 1 (q) < π , or if θ 1 …”

Section: Generalization To Higher Dimensionsmentioning

confidence: 99%

“…We recently presented in [1] a tight bound on the maximum number of vertices and facets in sums of three-dimensional polytopes, by showing that a bound on the number of vertices of a sum of r summands can be deduced from the number of vertices of the summands and the number of vertices of sums of each of the r 2 pairs of summands.…”

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confidence: 99%

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Maximal f-Vectors of Minkowski Sums of Large Numbers of Polytopes

Weibel

2011

Discrete Comput Geom

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It is known that in the Minkowski sum of r polytopes in dimension d, with r < d, the number of vertices of the sum can be as high as the product of the number of vertices in each summand. However, the number of vertices for sums of more polytopes was unknown so far.In this paper, we study sums of polytopes and prove a linear relation between the number of faces of a sum of r polytopes in dimension d, with r ≥ d, and the number of faces in the sums of less than d of the summand polytopes. We deduce from this result a bound on the maximum possible number of vertices of the Minkowski sum of any number of polytopes in any dimension. In particular, the linear relation implies that a sum of r polytopes in dimension d, where summands have n vertices in total, has less than n d−1 vertices, even when r ≥ d. Finally, we present a construction for any given number of vertices in summands and show that no other sum can achieve more vertices, establishing a precise tight bound.

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Section: Minkowski Sumsmentioning

confidence: 99%

“…Formally, for any points p and q of S d−1 that are not in U , we say that p is to the west of q if θ 1 (p) ∈ [θ 1 (q), θ 1 (q) + π] and θ 1 (q) < π , or if θ 1 …”

Section: Generalization To Higher Dimensionsmentioning

confidence: 99%

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Maximal f-Vectors of Minkowski Sums of Large Numbers of Polytopes

Weibel

2011

Discrete Comput Geom

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“…We call the intersection of this fan with the unit sphere the normal map of P . (This is called the gaussian map of P in [15]. Incidentally, the reading of [15] was our initial inspiration for attempting to disprove the Hirsch Conjecture via prismatoids.)…”

Section: Second Proof Of Theorem 31mentioning

confidence: 99%

“…(This is called the gaussian map of P in [15]. Incidentally, the reading of [15] was our initial inspiration for attempting to disprove the Hirsch Conjecture via prismatoids.) The normal map of a dpolytope P is a polyhedral complex of spherical polytopes decomposing S d−1 .…”

Section: Second Proof Of Theorem 31mentioning

confidence: 99%

A counterexample to the Hirsch Conjecture

2012

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The Hirsch Conjecture (1957) stated that the graph of a d-dimensional polytope with n facets cannot have (combinatorial) diameter greater than n − d. That is, any two vertices of the polytope can be connected by a path of at most n − d edges.This paper presents the first counterexample to the conjecture. Our polytope has dimension 43 and 86 facets. It is obtained from a 5-dimensional polytope with 48 facets that violates a certain generalization of the d-step conjecture of Klee and Walkup.

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The Maximum Number of Faces of the Minkowski Sum of Two Convex Polytopes

Karavelas

Tzanaki

2015

Discrete Comput Geom

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We derive tight expressions for the maximum number of k-faces, 0 ≤ k ≤ d − 1, of the Minkowski sum, P 1 ⊕ P 2 , of two d-dimensional convex polytopes P 1 and P 2 , as a function of the number of vertices of the polytopes.For even dimensions d ≥ 2, the maximum values are attained when P 1 and P 2 are cyclic dpolytopes with disjoint vertex sets. For odd dimensions d ≥ 3, the maximum values are attained when P 1 and P 2 are ⌊ d 2 ⌋-neighborly d-polytopes, whose vertex sets are chosen appropriately from two distinct d-dimensional moment-like curves.. This bound is attained when the polytopes P i are zonotopes, and their generating edges are in general position.Regarding bounds as a function of the number of vertices or facets of the summands, Fukuda and Weibel [5] have shown that, given two 3-polytopes P 1 and P 2 in E 3 , the number of k-faces of P 1 ⊕ P 2 , 0 ≤ k ≤ 2, is bounded from above as follows:where n j is the number of vertices of P j , j = 1, 2. Weibel [18] has also derived similar expressions in terms of the number of facets m j of P j , j = 1, 2, namely:All these bounds are tight. Fogel, Halperin and Weibel [3] have further generalized some of these bounds in the case of r summands. More precisely, they have shown that given r 3-polytopes

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On the Exact Maximum Complexity of Minkowski Sums of Polytopes

Abstract: We present a tight bound on the exact maximum complexity of Minkowski sums of polytopes in R 3 . 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 1≤i Show more

Cited by 21 publications

References 15 publications

Maximal f-Vectors of Minkowski Sums of Large Numbers of Polytopes

Maximal f-Vectors of Minkowski Sums of Large Numbers of Polytopes

A counterexample to the Hirsch Conjecture

The Maximum Number of Faces of the Minkowski Sum of Two Convex Polytopes

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