Directional curvature and medial axis

TL;DR

This paper introduces a notion of directional curvature to analyze the medial axis of definable sets, characterizing singularities without requiring smoothness and extending previous work to higher codimensions.

Contribution

It generalizes the concept of superquadraticity and provides a new criterion for understanding medial axis singularities in non-smooth, definable sets.

Findings

Established a criterion for medial axis singularities using directional curvature.

Extended the analysis of medial axes to higher codimensions.

Completed the study of the plane case for medial axis singularities.

Abstract

The medial axis $M_X$ of a closed set $X\subset \mathbb{R}^n$ is the set of points from the ambient space that admit more than one closest point in $X$. We study the problem of reaching the singularities, i.e. of characterising the points of the set $\overline{M_X}\cap X$. In order to tame the geometry, we assume that $X$ is definable in a polynomially bounded structure and obtain a general criterion based on a generalisation of the notion of superquadraticity previously introduced by Birbrair and Denkowski for $C^1$-smooth hypersurfaces and extended to any codimension by Bia{\l}o\.zyt. We do not require any smoothness as we achieve our goal by introducing a notion of directional curvature in some naturally chosen camber directions. This allows us in particular to complete the study of the plane case.