Papers by Christophe Weibel
On 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 polyto... more 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 face lattice of the sum in terms of the face
Minimum perimeter convex hull of imprecise points in convex regions
Imprecise points are points in R2 whose exact location is unknown. For each point, we only know i... more Imprecise points are points in R2 whose exact location is unknown. For each point, we only know it is contained in a region of R2, which is called the uncertainty region of the point. The research we present in this video focuses on the problem of finding the minimum perimeter convex hull of a set of imprecise points, where each
Computing Research Repository, 2010
We give an algorithm to route a multicommodity flow in a planar graph $G$ with congestion $O(\log... more We give an algorithm to route a multicommodity flow in a planar graph $G$ with congestion $O(\log k)$, where $k$ is the maximum number of terminals on the boundary of a face, when each demand edge lie on a face of $G$. We also show that our specific method cannot achieve a substantially better congestion.
This is a short paper on dieren t proofs for special cases of a conjecture about Minkowski sums o... more This is a short paper on dieren t proofs for special cases of a conjecture about Minkowski sums of polytopes.
Let G=(V,E) be a supply graph and H=(V,F) a demand graph defined on the same set of vertices. An ... more 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
On the Computation of 3D Visibility Skeletons
Computing and Combinatorics, 2010
The 3D visibility skeleton is a data structure that encodes the global visibility information of ... more The 3D visibility skeleton is a data structure that encodes the global visibility information of a set of 3D objects. While it is useful in answering global visibility queries, its large size often limits its practical use. In this paper, we address this issue by proposing a subset of the visibility skeleton, which is empirically about 25% to 50% of
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 ... more 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.
Minimum perimeter convex hull of imprecise points in convex regions
Proceedings of the 27th annual ACM symposium on Computational geometry - SoCG '11, 2011
ABSTRACT Imprecise points are points in R2 whose exact location is unknown. For each point, we on... more ABSTRACT Imprecise points are points in R2 whose exact location is unknown. For each point, we only know it is contained in a region of R2, which is called the uncertainty region of the point. The research we present in this video focuses on the problem of finding the ...
Lecture Notes in Computer Science, 2008
The 3D visibility skeleton is a data structure used to encode global visibility information about... more 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 k convex polytopes with n edges in total, the worst case size complexity of this data structure is Θ(n 2 k 2 ) [Brönnimann et al. 07]; whereas for k uniformly distributed unit spheres, the expected size is Θ(k) [Devillers et al. 03].
We present a tight bound on the exact maximum complexity of Minkowski sums of polytopes in R 3 . ... more 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
The objective of this paper is to present two types of results on Minkowski sums of convex polyto... more 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 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.
The objective of this paper is to study a special family of Minkowski sums, that is of polytopes ... more The objective of this paper is to study a special family of Minkowski sums, that is of polytopes relatively in general position. We show that the maximum number of faces in the sum can be attained by this family. We present a new linear equation that is satisfied by f -vectors of the sum and the summands. We study some of the implications of this equation.
Lecture Notes in Computer Science, 2011
For a set S of n lines labeled from 1 to n, we say that S supports an n-vertex planar graph G if ... more For a set S of n lines labeled from 1 to n, we say that S supports an n-vertex planar graph G if for every labeling from 1 to n of its vertices, G has a straight-line crossing-free drawing with each vertex drawn as a point on its associated line. It is known from previous work [4] that no set of n parallel lines supports all n-vertex planar graphs. We show that intersecting lines, even if they intersect at a common point, are more "powerful" than a set of parallel lines. In particular, we prove that every such set of lines supports outerpaths, lobsters, and squids, none of which are supported by any set of parallel lines. On the negative side, we prove that no set of n lines that intersect in a common point supports all n-vertex planar graphs. Finally, we show that there exists a set of n lines in general position that does not support all n-vertex planar graphs.
Discrete & Computational Geometry, 2009
We present a tight bound on the exact maximum complexity of Minkowski sums of polytopes in R 3 . ... more 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
Collection of Abstracts of the 23rd …, 2007
Any face of a Minkowski sum of polytopes can be decomposed uniquely into a sum of faces of the su... more Any face of a Minkowski sum of polytopes can be decomposed uniquely into a sum of faces of the summands. We will say that the decomposition is exact when the dimension of the sum is equal to the sum of the dimensions of the summands. When all facets have an exact ...
Proceedings of the twenty-third …, 2007
We present a tight bound on the exact maximum complexity of Minkowski sums of convex polyhedra in... more 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.
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Papers by Christophe Weibel