Curvatures for unions of WDC sets

Abstract

We prove the existence of the curvature measures for a class of scriptUWDC${\mathcal {U}}_{{\rm WDC}}$ sets, which is a direct generalisation of scriptUPR${\mathcal {U}}_{\rm {P\! R}}$ sets studied by Rataj and Zähle. Moreover, we provide a simple characterisation of scriptUWDC${\mathcal {U}}_{{\rm WDC}}$ sets in double-struckR2$\mathbb {R}^2$ and prove that in double-struckR2$\mathbb {R}^2$, the class of scriptUWDC${\mathcal {U}}_{{\rm WDC}}$ sets contains essentially all classes of sets known to admit curvature measures.