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<p>While there is extensive literature on approximation, deterministic as well as random, of general convex bodies <i>K</i> in the symmetric difference metric, or other metrics arising from intrinsic volumes, very little is known for corresponding random results in the Hausdorff distance when the approximant <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3186_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(K_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> is given by the convex hull of <i>n</i> independent random points chosen uniformly on the boundary or in the interior of <i>K</i>. When <i>K</i> is a polygon and the points are chosen on its boundary, we determine the exact limiting behavior of the expected Hausdorff distance. From this we derive the behavior of the asymptotic constant for a regular polygon in the number of vertices.</p>

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Random approximation of convex bodies in Hausdorff distance

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年6月7日

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  • Joscha Prochno,
  • Carsten Schütt,
  • Mathias Sonnleitner,
  • Elisabeth M. Werner

摘要

While there is extensive literature on approximation, deterministic as well as random, of general convex bodies K in the symmetric difference metric, or other metrics arising from intrinsic volumes, very little is known for corresponding random results in the Hausdorff distance when the approximant \(K_n\) K n is given by the convex hull of n independent random points chosen uniformly on the boundary or in the interior of K. When K is a polygon and the points are chosen on its boundary, we determine the exact limiting behavior of the expected Hausdorff distance. From this we derive the behavior of the asymptotic constant for a regular polygon in the number of vertices.