Medial Axis in Pseudo-Euclidean Spaces

TL;DR

This work extends the Euclidean concept of the medial axis to pseudo-Euclidean spaces $(\mathbb{R}^n,Q)$ by defining the distance function $\rho(a,X)=\inf_{x\in X} Q(x-a)$ and the closest-point map $m(a)$. It develops the associated structure $\mathcal{N}(a)$, proves regularity and definability results (compactness, upper semicontinuity, and differentiability of $\rho$ off the medial axis) under acausal and $L$-pseudo-Lipschitz hypotheses, and introduces a local central set $C_X$ with the containment $M_X\subset C_X\subset \overline{M_X}$. A pseudo-Euclidean Nash-type lemma shows $\overline{M_X}$ has no intersection with the 2nd-regular part, and the framework is analyzed under Kuratowski limits of growing families of sets. For hypersurfaces, the authors establish local univalence of the closest-point map on suitable slices, yielding a local homeomorphism from parts of the skeleton to the regular part of the hypersurface and connecting the medial axis to a local spherical structure around $m(a)$.

Abstract

We investigate the notion of the medial axis for pseudo-Euclidean spaces. For most of the article, we follow the path of Birbrair and Denkowski's article "Medial Axis and Singularities", checking its feasibility in the new context.

Theorems & Definitions (35)

• Definition 1.1
• Theorem 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Theorem 3.4
• proof
• Theorem 3.5