Curvatures for unions of WDC sets

TL;DR

This paper establishes the existence of curvature measures for a broad class of sets called al U_{ m WDC}, generalizing previous classes and providing characterizations in al R^2, showing it encompasses most known sets with curvature measures.

Contribution

It introduces al U_{ m WDC} sets, proves curvature measures exist for them, and characterizes these sets in al R^2, unifying known classes.

Findings

Curvature measures exist for al U_{ m WDC} sets.

In al R^2, al U_{ m WDC} includes all known sets with curvature measures.

Provides a simple characterization of al U_{ m WDC} sets in al R^2.

Abstract

We prove the existence of the curvature measures for a class of $\cal U_{\rm WDC}$ sets, which is a direct generalization of $\cal U_{\rm PR}$ sets studied by Rataj and Z\"ahle. Moreover, we provide a simple characterisation of $\cal U_{\rm WDC}$ sets in $\mathbb R^2$ and prove that in $\mathbb R^2$ the class of $\cal U_{\rm WDC}$ sets contains essentially all classes of sets known to admit curvature measures.