On the zone of the boundary of a convex body

TL;DR

This paper investigates the complexity of the boundary zone of a convex set within hyperplane arrangements, showing the outer boundary's complexity is linear in the number of hyperplanes, which refines previous bounds.

Contribution

It proves that the outer part of the zone has complexity O(n^{d-1}), improving understanding of boundary complexities in hyperplane arrangements.

Findings

Outer zone complexity is O(n^{d-1})

Inner zone complexity remains an open problem

Refines previous known bounds on zone complexity

Abstract

We consider an arrangement $\A$ of $n$ hyperplanes in $\R^d$ and the zone $\Z$ in $\A$ of the boundary of an arbitrary convex set in $\R^d$ in such an arrangement. We show that, whereas the combinatorial complexity of $\Z$ is known only to be $O<n^{d-1}\log n>$ \cite{APS}, the outer part of the zone has complexity $O<n^{d-1}>$ (without the logarithmic factor). Whether this bound also holds for the complexity of the inner part of the zone is still an open question (even for $d=2$).