On volume and surface area of parallel sets. II. Surface measures and (non-)differentiability of the volume

TL;DR

This paper investigates the differentiability of the volume function of compact sets in Euclidean space, showing convergence of surface area measures at differentiability points and characterizing non-differentiability sets in low dimensions.

Contribution

It establishes the weak convergence of surface area measures at differentiability points and fully characterizes non-differentiability sets for dimensions one and two.

Findings

Surface area measures converge weakly at differentiability points.

Characterization of non-differentiability sets in dimensions 1 and 2.

Conditions for sets of radii where volume function is non-differentiable.

Abstract

We prove that at differentiability points $r_0>0$ of the volume function of a compact set $A\subset {\mathbb R}^d$ (associating to $r$ the volume of the $r$-parallel set of $A$), the surface area measures of $r$-parallel sets of $A$ converge weakly to the surface area measure of the $r_0$-parallel set as $r\to r_0$. We further study the question which sets of parallel radii can occur as sets of non-differentiability points of the volume function of some compact set. We provide a full characterization for dimensions $d=1$ and $2$.