Reverse isoperimetric inequalities for parallel sets

TL;DR

This paper establishes upper bounds on the surface area to volume ratio for $r$-parallel sets in Euclidean space, with implications for Gaussian surface measures, revealing optimal cases and bounds.

Contribution

It introduces reverse isoperimetric inequalities for parallel sets, providing new bounds on surface area measures and their Gaussian counterparts.

Findings

Surface area to volume ratio bounded by d/r, equality for single points.

Gaussian surface area measure bounded by 18d times the maximum of sqrt(d) and 1/r.

Existence of a 1-parallel set with Gaussian surface area measure at least 0.28·d^{1/4}.

Abstract

We consider the family of $r$-parallel sets in $\mathbb{R}^d$, that is sets of the form $A_r=A+rB_2^n$, where $B_2^n$ is the unit Euclidean ball and $A$ is an arbitrary Borel set. We show that the ratio between the upper surface area measure of an $r$-parallel set and its volume is upper bounded by $d/r$. Equality is achieved for $A$ being a single point. As a consequence of our main result we show that the Gaussian upper surface area measure of an $r$-parallel set is upper bounded by $18d \max(\sqrt{d},r^{-1})$. Moreover, we observe that there exists a $1$-parallel set with Gaussian surface area measure at least $0.28 \cdot d^{1/4}$.