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Triangulations: Structures for Algorithms and Applications
Triangulations appear everywhere, from volume computations and meshing to algebra and topology. This book studies the subdivisions and triangulations of polyhedral regions and point sets and presents…
A counterexample to the Hirsch conjecture
- F. Santos
- 14 June 2010
This paper presents the rst counterexample to the Hirsch Conjecture, obtained from a 5-dimensional polytope with 48 facets that violates a certain generalization of the d-step conjecture of Klee and Walkup.
Higher Lawrence configurations
Expansive Motions and the Polytope of Pointed Pseudo-Triangulations
We introduce the polytope of pointed pseudo-triangulations of a point set in the plane, defined as the polytope of infinitesimal expansive motions of the points subject to certain constraints on the…
The Polytope of All Triangulations of a Point Configuration
We study the convex hull P A of the 0-1 incidence vectors of all triangulations of a point con guration A. This was called the universal polytope in . The a ne span of P A is described in terms of…
Pseudo-Triangulations - a Survey
A pseudo-triangle is a simple polygon with three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region into pseudo-triangles. Pseudo-triangulations appear as data…
The Cayley Trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings
Abstract.In 1994, Sturmfels gave a polyhedral version of the Cayley Trick of elimination theory: he established an order-preserving bijection between the posets of coherent mixed subdivisions of a…
Triangulations of oriented matroids
- F. Santos
Introduction Preliminaries on oriented matroids Triangulations of oriented matroids Duality between triangulations and extensions Subdivisions of Lawrence polytopes Lifting triangulations…
Asymptotically Efficient Triangulations of the d-Cube
The method has a computational part, where it is shown how to triangulate the product P \times Q efficiently (i.e., with few simplices) starting with a given triangulation of $Q$.
Planar minimally rigid graphs and pseudo-triangulations
This paper proves that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints, and provides the first algorithmically effective result on graph embeddeddings with oriented matroid constraints other than convexity of faces.