# On the structure of WDC sets

## TL;DR

This paper investigates the structure and regularity of WDC sets, which are sublevel sets of DC functions, revealing their boundary properties, decompositions, and characterizations in various dimensions.

## Contribution

It provides new regularity results, boundary decompositions, and characterizations of WDC sets, extending understanding beyond sets with positive reach.

## Key findings

- Boundaries of compact WDC sets can be covered by finitely many DC surfaces.
- Any compact WDC set can be decomposed into a DC manifold and lower-dimensional DC surfaces.
- Full characterization of locally WDC sets in the plane.

## Abstract

WDC sets in ${\mathbb R}^d$ were recently defined as sublevel sets of DC functions (differences of convex functions) at weakly regular values. They form a natural and substantial generalization of sets with positive reach and still admit the definition of curvature measures. Using results on singularities of convex functions, we obtain regularity results on the boundaries of WDC sets. In particular, the boundary of a compact WDC set can be covered by finitely many DC surfaces. More generally, we prove that any compact WDC set $M$ of topological dimension $k\leq d$ can be decomposed into the union of two sets, one of them being a $k$-dimensional DC manifold open in $M$, and the other can be covered by finitely many DC surfaces of dimension $k-1$. We also characterize locally WDC sets among closed Lipschitz domains and among lower-dimensional Lipschitz manifolds. Finally, we find a full characterization of locally WDC sets in the plane.