Christophe Weibel

f-Vectors of Minkowski Additions of Convex Polytopes

The objective of this paper is to present two types of results on Minkowski sums of convex polytopes. The first is about a special class of polytopes we call perfectly centered and the combinatorial properties of the Minkowski sum with their own dual. In particular, we have a characterization of the face lattice of the sum in terms of the face lattice of a given perfectly centered polytope. Exact face counting formulas are then obtained for perfectly centered simplices and hypercubes. The second type of results concerns tight upper bounds for the f-vectors of Minkowski sums of several polytopes.

On the Exact Maximum Complexity of Minkowski Sums of Polytopes

AbstractWe 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 m1,m2,…,mk facets, respectively, is bounded from above by

When the cut condition is enough: a complete characterization for multiflow problems in series-parallel networks

\sum_{1\leq i<j\leq k}(2m_{i}-5)(2m_{j}-5)+\sum_{1\leq i\leq k}m_{i}+\binom{k}{2}

On the Size of the 3D Visibility Skeleton: Experimental Results

. Given k positive integers m1,m2,…,mk, we describe how to construct k polytopes with corresponding number of facets, such that the number of facets of their Minkowski sum is exactly

On the exact maximum complexity of Minkowski sums of convex polyhedra

\sum_{1\leq i<j\leq k}(2m_{i}-5)(2m_{j}-5)+\sum_{1\leq i\leq k}m_{i}+\binom{k}{2}

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

. When k=2, for example, the expression above reduces to 4m1m2−9m1−9m2+26.

A linear equation for Minkowski sums of polytopes relatively in general position

Let G=(V,E) be a supply graph and H=(V,F) a demand graph defined on the same set of vertices. An assignment of capacities to the edges of G and demands to the edges of H is said to satisfy the cut condition if for any cut in the graph, the total demand crossing the cut is no more than the total capacity crossing it. The pair (G,H) is called cut-sufficient if for any assignment of capacities and demands that satisfy the cut condition, there is a multiflow routing the demands defined on

Connectivity Graphs of Uncertainty Regions

H

Minimum perimeter convex hull of imprecise points in convex regions

within the network with capacities defined on G. We prove a previous conjecture, which states that when the supply graph G is series-parallel, the pair (G,H) is cut-sufficient if and only if (G,H) does not contain an odd spindle as a minor; that is, if it is impossible to contract edges of G and delete edges of G and H so that G becomes the complete bipartite graph K2,p, with p ≥ 3 odd, and H is composed of a cycle connecting the p vertices of degree 2, and an edge connecting the two vertices of degree p. We further prove that if the instance is Eulerian --- that is, the demands and capacities are integers and the total of demands and capacities incident to each vertex is even --- then the multiflow problem has an integral solution. We provide a polynomial-time algorithm to find an integral solution in this case. In order to prove these results, we formulate properties of tight cuts (cuts for which the cut condition inequality is tight) in cut-sufficient pairs. We believe these properties might be useful in extending our results to planar graphs.

On the computation of 3D visibility skeletons

The 3D visibility skeleton is a data structure used to encode global visibility information about a set of objects. Previous theoretical results have shown that for kconvex polytopes with nedges in total, the worst case size complexity of this data structure is i¾?(n2k2) [Bronnimann et al. 07]; whereas for kuniformly distributed unit spheres, the expected size is i¾?(k) [Devillers et al. 03]. In this paper, we study the size of the visibility skeleton experimentally. Our results indicate that the size of the 3D visibility skeleton, in our setting, is