Concavity properties of extensions of the parallel volume

TL;DR

This paper investigates concavity properties of extended outer parallel volumes, including general measures and functional versions, broadening classical geometric measure theory.

Contribution

It introduces new concavity results for generalized measures and functional extensions of outer parallel sets, expanding the scope of classical geometric inequalities.

Findings

Concavity properties hold for measures beyond Lebesgue measure.

Functional versions of outer parallel sets exhibit similar concavity properties.

Results generalize classical geometric measure inequalities.

Abstract

In this paper we establish concavity properties of two extensions of the classical notion of the outer parallel volume. On the one hand, we replace the Lebesgue measure by more general measures. On the other hand, we consider a functional version of the outer parallel sets.