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.
Categories: math.AP