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<p>The medial axis of a closed set collects all the points of the ambient space with nonunique distance realiser in a set. This paper studies which of the set’s singular points are reached by the medial axis. The investigation of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10851_2025_1237_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathscr {C}^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="script">C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> smoothness points extends earlier results of Birbrair and Denkowski of superquadratic points to dimensions greater than two and to general o-minimal structures. The collection of points where the set fails to be <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10851_2025_1237_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathscr {C}^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="script">C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> smooth is studied from the perspective of its tangent cone and the biggest submanifold found in the set.</p>

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On the Singular Points Approached by the Medial Axis

  • 期刊论文

  • 发表日期：

    2025年4月9日

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  • Adam Bialozyt

摘要

The medial axis of a closed set collects all the points of the ambient space with nonunique distance realiser in a set. This paper studies which of the set’s singular points are reached by the medial axis. The investigation of \(\mathscr {C}^1\) C 1 smoothness points extends earlier results of Birbrair and Denkowski of superquadratic points to dimensions greater than two and to general o-minimal structures. The collection of points where the set fails to be \(\mathscr {C}^1\) C 1 smooth is studied from the perspective of its tangent cone and the biggest submanifold found in the set.