# Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension

Discrete & Computational Geometry, May 2013

For a polyhedron $$P$$ let $$B(P)$$ denote the polytopal complex that is formed by all bounded faces of $$P$$. If $$P$$ is the intersection of $$n$$ halfspaces in $$\mathbb R ^D$$, but the maximum dimension $$d$$ of any face in $$B(P)$$ is much smaller, we show that the combinatorial complexity of $$P$$ cannot be too high; in particular, that it is independent of $$D$$. We show that the number of vertices of $$P$$ 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 $$O(n^d)$$. We show that for certain specific values of $$d$$ and $$D$$, our bounds are tight. 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 number of linear programs that is polynomial in $$n$$.

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David Eppstein, Maarten Löffler. Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension, Discrete & Computational Geometry, 2013, 1-21, DOI: 10.1007/s00454-013-9503-3