Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension
Computational Geometry
2013-07-30 v1 Discrete Mathematics
Abstract
We study the combinatorial complexity of D-dimensional polyhedra defined as the intersection of n halfspaces, with the property that the highest dimension of any bounded face is much smaller than D. We show that, if d is the maximum dimension of a bounded face, then the number of vertices of the polyhedron is O(n^d) and the total number of bounded faces of the polyhedron is O(n^d^2). For inputs in general position the number of bounded faces is O(n^d). For any fixed d, we show how to compute the set of all vertices, how to determine the maximum dimension of a bounded face of the polyhedron, and how to compute the set of bounded faces in polynomial time, by solving a polynomial number of linear programs.
Keywords
Cite
@article{arxiv.1103.2575,
title = {Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension},
author = {David Eppstein and Maarten Löffler},
journal= {arXiv preprint arXiv:1103.2575},
year = {2013}
}