Minimally singular functions and the rigidity problem for Steiner's perimeter inequality

TL;DR

This paper introduces minimally singular functions in BV spaces, characterizes them geometrically, and applies this to determine when equality cases in Steiner's perimeter inequality are rigid, meaning only vertical translations are extremals.

Contribution

It defines minimally singular functions in BV, provides a geometric characterization using a new pseudometric, and applies this to characterize rigidity in Steiner's perimeter inequality.

Findings

Characterization of minimally singular functions via singular vertical distance.

Geometric criteria for rigidity in Steiner's perimeter inequality.

Rigidity holds only for vertical translations of Steiner symmetric sets.

Abstract

Let $n\geq 1$, and let $\Omega\subset \mathbb{R}^n$ be an open and connected set with finite Lebesgue measure. Among functions of bounded variation in $\Omega$ we introduce the class of \emph{minimally singular} functions. Inspired by the original theory of Vol'pert of one-dimensional restrictions of $BV$ functions, we provide a geometric characterization for this class of functions via the introduction of a pseudometric that we call \emph{singular vertical distance}. As an application, we present a characterization result for \emph{rigidity} of equality cases for Steiner's perimeter inequality. By \emph{rigidity} we mean that the only extremals for Steiner's perimeter inequality are vertical translations of the Steiner symmetric set.