Voronoi diagrams of algebraic varieties under polyhedral norms

TL;DR

This paper investigates Voronoi diagrams of algebraic varieties under polyhedral norms, providing bounds on cell dimensions and counting full-dimensional cells, with applications to polyhedral Wasserstein distances.

Contribution

It introduces bounds on Voronoi cell dimensions and counts full-dimensional cells for algebraic varieties under polyhedral norms, advancing geometric understanding.

Findings

Bounds on Voronoi cell dimensions established

Number of full-dimensional cells counted for algebraic varieties

Application to polyhedral Wasserstein distance analyzed

Abstract

We study Voronoi diagrams of manifolds and varieties with respect to polyhedral norms. We provide upper and lower bounds on the dimensions of Voronoi cells. For algebraic varieties, we count their full-dimensional Voronoi cells. As an application, we consider the polyhedral Wasserstein distance between discrete probability distributions.