Polyhedral combinatorics of bisectors

TL;DR

This paper explores the combinatorial structure of bisectors under polyhedral norms, extending previous work on tropical distances, and provides explicit descriptions and bounds for various norms in different dimensions.

Contribution

It introduces the concept of bisection cones and fans for arbitrary polyhedral norms, establishing their existence in two dimensions and analyzing specific norms like  and .

Findings

Bisection fan exists for all polyhedral norms in two dimensions.

Explicit bisection fans are determined for  and  norms.

Bounds on the combinatorial complexity of bisectors are derived.

Abstract

For any polyhedral norm, the bisector of two points is a polyhedral complex. We study combinatorial aspects of this complex. We investigate the sensitivity of the presence of labelled maximal cells in the bisector relative to the position of the two points. We thereby extend work of Criado, Joswig and Santos (2022) who showed that for the tropical distance function the presence of maximal cells is encoded by a polyhedral fan, the bisection fan. We initiate the study of bisection cones and bisection fans with respect to arbitrary polyhedral norms. In particular, we show that the bisection fan always exists for polyhedral norms in two dimensions. Furthermore, we determine the bisection fan of the $\ell_{1}$-norm and the $\ell_{\infty}$-norm as well as the discrete Wasserstein distance in arbitrary dimensions. Intricate combinatorial structures, such as the resonance arrangement, make their appearance. We apply our results to obtain bounds on the combinatorial complexity of the bisectors.