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arXiv · 2411.17633

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

Matteo Perugini
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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.

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Matteo Perugini. 2024-11-26. Minimally singular functions and the rigidity problem for Steiner's perimeter inequality. https://arxiv.org/abs/2411.17633

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