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<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n\ge 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Omega \subset \mathbb {R}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> be an open and connected set with finite Lebesgue measure. Among functions of bounded variation in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\Omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> we introduce the class of <i>minimally singular</i> functions. Inspired by the original theory of Vol’pert of one-dimensional restrictions of <i>BV</i> functions, we provide a geometric characterization for this class of functions via the introduction of a pseudometric that we call <i>singular vertical distance</i>. As an application, we present a characterization result for <i>rigidity</i> of equality cases for Steiner’s perimeter inequality. By <i>rigidity</i> we mean that the only extremals for Steiner’s perimeter inequality are vertical translations of the Steiner symmetric set.</p>

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

  • 期刊论文

  • 发表日期：

    2026年4月1日

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  • Matteo Perugini

摘要

Let \(n\ge 1\) n ≥ 1 , and let \(\Omega \subset \mathbb {R}^n\) Ω ⊂ R n be an open and connected set with finite Lebesgue measure. Among functions of bounded variation in \(\Omega \) Ω we introduce the class of 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 singular vertical distance. As an application, we present a characterization result for rigidity of equality cases for Steiner’s perimeter inequality. By rigidity we mean that the only extremals for Steiner’s perimeter inequality are vertical translations of the Steiner symmetric set.