Algebraic Distance Optimization in Polyhedral Norms

TL;DR

This paper studies the geometry of distance minimization to algebraic varieties under polyhedral norms, introducing a stratification of the variety based on Voronoi cell dimensions and describing the medial axis algebraically.

Contribution

It provides a novel decomposition of algebraic varieties into semialgebraic strata based on Voronoi cone dimensions under polyhedral norms, and characterizes the medial axis algebraically.

Findings

Decomposition of varieties into semialgebraic strata based on Voronoi cone dimensions.

Proof that the decomposition forms a stratification.

Algebraic description of the medial axis.

Abstract

We consider the distance minimization problem to a real algebraic variety $X \subseteq \RR^n$ when the metric is induced by a polyhedral norm. Each point in the variety has a Voronoi cell whose geometry depends on the normal space at the point and the inner normal fan of the polyhedral ball. For codimension-one varieties, we decompose $X$ into sets of points whose Voronoi cones have the same dimension, which is the expected dimension of their Voronoi cell. We prove that this decomposition is a stratification of $X$ and that each strata is a semialgebraic set. We conclude by giving an algebraic description of the medial axis, which is the locus of points whose minimal distance to $X$ is achieved at more than one point on $X$.