# Remarks on WDC sets

## TL;DR

This paper investigates WDC sets in the plane, showing that their distance functions serve as DC auras, which leads to new insights about their structure and measurability within the space of compact sets.

## Contribution

It proves that the distance function is a DC aura for WDC sets in \\mathbb{R}^2, establishing that locally WDC sets are WDC and that compact WDC sets form a Borel subset.

## Key findings

- Distance function acts as a DC aura for WDC sets in \\mathbb{R}^2
- Locally WDC sets in \\mathbb{R}^2 are WDC sets
- Compact WDC sets form a Borel subset of all compact sets

## Abstract

We study WDC sets, which form a substantial generalization of sets with positive reach and still admit the definition of curvature measures. Main results concern WDC sets $A\subset \mathbb{R}^2$. We prove that, for such $A$, the distance function $d_A= {\rm dist}(\cdot,A)$ is a `DC aura' for $A$, which implies that each locally WDC set in $\mathbb{R}^2$ is a WDC set. An another consequence is that compact WDC subsets of $\mathbb{R}^2$ form a Borel subset of the space of all compact sets.