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arXiv · 2604.26490

Directional curvature and medial axis

Adam Białożyt·Dominik Bysiewicz·Maciej P. Denkowski
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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.

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Adam Białożyt, Dominik Bysiewicz, Maciej P. Denkowski. 2026-04-29. Directional curvature and medial axis. https://arxiv.org/abs/2604.26490

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