Discrete & Computational Geometry
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On the coordinatization of oriented matroids
Discrete & Computational Geometry - Tập 1 - Trang 293-306 - 1986
Several important and hard realizability problems of combinatorial geometry can be reduced to the realizability problem of oriented matroids. In this paper we describe a method to find a coordinatization for a large class of realizable cases. This algorithm has been used successfully to decide several geometric realizability problems. It is shown that all realizations found by our algorithm fulfil......
Continuous and Discrete Radius Functions on Voronoi Tessellations and Delaunay Mosaics
Discrete & Computational Geometry - Tập 67 - Trang 811-842 - 2022
The Voronoi tessellation in
$${{{\mathbb {R}}}}^d$$
is defined by locally minimizing the power distance to given weighted points. Symmetrically, the Delaunay mosaic can be defined by locally maximizing the negative power distance to other such points. We prove that the average of the two piecewise......
On Open and Closed Convex Codes
Discrete & Computational Geometry - Tập 61 Số 2 - Trang 247-270 - 2019
Ehrhart Polynomials of Matroid Polytopes and Polymatroids
Discrete & Computational Geometry - Tập 42 - Trang 670-702 - 2008
We investigate properties of Ehrhart polynomials for matroid polytopes, independence matroid polytopes, and polymatroids. In the first half of the paper we prove that, for fixed rank, Ehrhart polynomials of matroid polytopes and polymatroids are computable in polynomial time. The proof relies on the geometry of these polytopes as well as a new refined analysis of the evaluation of Todd polynomials......
$$\varepsilon $$ -Isometric Dimension Reduction for Incompressible Subsets of $$\ell _p$$
Discrete & Computational Geometry - - Trang 1-17 - 2023
Fix
$$p\in [1,\infty )$$
,
$$K\in (0,\infty )$$
, and a probability measure
$$\mu $$
. We prove that for every
......
Topological Optimization with Big Steps
Discrete & Computational Geometry - - Trang 1-35 - 2024
Using persistent homology to guide optimization has emerged as a novel application of topological data analysis. Existing methods treat persistence calculation as a black box and backpropagate gradients only onto the simplices involved in particular pairs. We show how the cycles and chains used in the persistence calculation can be used to prescribe gradients to larger subsets of the domain. In pa......
Pre-Triangulations and Liftable Complexes
Discrete & Computational Geometry - Tập 38 - Trang 701-725 - 2007
We introduce the concept of pre-triangulations, a relaxation of triangulations that goes beyond the frequently used concept of pseudo-triangulations. Pre-triangulations turn out to be more natural than pseudo-triangulations in certain cases. We show that pre-triangulations arise in three different contexts: In the characterization of polygonal complexes that are liftable to three-space in a stron......
Pairwise Balanced Designs with Prescribed Minimum Dimension
Discrete & Computational Geometry - Tập 51 - Trang 485-494 - 2013
The dimension of a linear space is the maximum positive integer d such that any d of its points generate a proper subspace. For a set K of integers at least two, recall that a pairwise balanced design
$\operatorname{PBD}(v,K)$
is a linear space on v points whose lines (or blocks) have sizes belonging to ......
Explicit Constructions of Centrally Symmetric $$k$$ -Neighborly Polytopes and Large Strictly Antipodal Sets
Discrete & Computational Geometry - Tập 49 Số 3 - Trang 429-443 - 2013
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