On the L p Minkowski problem for polytopes (original) (raw)

2005

This note summarizes recent results from computational geometry which determine complexity of computing Minkowski sum of k convex polytopes in R d , which are represented either in terms of facets or in terms of vertices. In particular, it is pointed out for which cases there exists an algorithm which runs in polynomial time. The note is based on papers of Gritzmann and Sturmfels [6] and Komei Fukuda . An algorithm which aims at reducing the complexity of obtaining minimal representation of polytopes given by a set of inequalities is presented as well.

Some convergence properties of Minkowski functionals given by polytopes

2016

In this work we investigate the behavior of the Minkowski Functionals admitted by a sequence of sets which converge to the unit ball ‘from the inside’. We begin in R2 and use this example to build intuition as we extend to the more general Rn case. We prove, in the penultimate chapter, that convergence ‘from the inside’ in this setting is equivalent to two other characterizations of the convergence: a geometric characterization which has to do with the sizes of the faces of each polytope in the sequence converging to zero, and the convergence of the Minkowski functionals defined on the approximating sets to the Euclidean Norm. In the last chapter we explore how we can extend our results to infinite dimensional vector spaces by changing our definition of polytope in that setting, the outlook is bleak. SOME CONVERGENCE PROPERTIES OF MINKOWSKI FUNCTIONALS GIVEN BY POLYTOPES A Thesis Submitted in Partial Fulfillment of the Requirement for the Degree Master of Arts Jesse Moeller Universi...

On the Exact Maximum Complexity of Minkowski Sums of Polytopes

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 sum1leqi<jleqk(2mi−5)(2mj−5)+sum1leqileqkmi+binomk2\sum_{1\leq i<j\leq k}(2m_{i}-5)(2m_{j}-5)+\sum_{1\leq i\leq k}m_{i}+\binom{k}{2}sum1leqi<jleqk(2mi−5)(2mj−5)+sum1leqileqkmi+binomk2 . 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 sum1leqi<jleqk(2mi−5)(2mj−5)+sum1leqileqkmi+binomk2\sum_{1\leq i<j\leq k}(2m_{i}-5)(2m_{j}-5)+\sum_{1\leq i\leq k}m_{i}+\binom{k}{2}sum1leqi<jleqk(2mi−5)(2mj−5)+sum1leqileqkmi+binomk2 . When k=2, for example, the expression above reduces to 4m 1m 2−9m 1−9m 2+26.

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

Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, 2012

We derive tight expressions for the maximum values of the 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.

On hyperplanes and polytopes

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.

Polytopes: Abstract, Convex and Computational

Polytopes: Abstract, Convex and Computational, 1994

Convex and Computational NATO ASI Series Advanced Science Institutes Series A Series presenting the results of activities sponsored by the NATO Science Committee, which aims at the dissemination of advanced scientific and technological knowledge, with a view to strengthening links between scientific communities.

Ball polytopes and the V�zsonyi problem

Acta Math Hung, 2010

Let V be a finite set of points in Euclidean d-space (d >= 2). The intersection of all unit balls B(v,1) centered at v, where v ranges over V, henceforth denoted by B(V) is the ball polytope associated with V. Note that B(V) is non-empty iff the circumradius of V is <= 1. After some preparatory discussion on spherical convexity and spindle convexity, the paper focuses on two central themes. [a] Define the boundary complex of B(V) (assuming it is non-empty, of course), i.e., define its vertices, edges and facets in dimension 3 (in dimension 2 this complex is just a circuit), and investigate its basic properties. [b] Apply results of this investigation to characterize finite sets of diameter 1 in (Euclidean) 3-space for which the diameter is attained a maximal number of times as a segment (of length 1) with both endpoints in V. A basic result for such a characterization goes back to Grunbaum, Heppes and Straszewicz, who proved independently that the diameter of V is attained at most 2|V|-2 times, thus affirming a conjecture of Vazsonyi from circa 1935. Call V extremal if its diameter is attained this maximal number (2|V|-2) of times. We extend the aforementioned basic result by showing that V is extremal iff V coincides with the set of vertices of its ball polytope B(V) and show that in this case the boundary complex of B(V) is self-dual in some strong sense. For the sake of priority we mention that, in the present form (except for a few changes in the footnotes), the paper was submitted to a journal already in February 1, 2008.

Problems on polytopes, their groups, and realizations

Periodica Mathematica Hungarica, 2006

The paper gives a collection of open problems on abstract polytopes that were either presented at the Polytopes Day in Calgary or motivated by discussions at the preceding Workshop on Convex and Abstract Polytopes at the Banff International Research Station in May 2006.

The maximum number of faces of the minkowski sum of three 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.