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<p>We consider a probability model in which the hull of a sample of i.i.d. uniform random points from a convex disc <i>K</i> is formed by the intersection of all translates of another suitable fixed convex disc <i>L</i> that contain the sample. Such an object is called a random <i>L</i>-polygon in <i>K</i>. We assume that both <i>K</i> and <i>L</i> have <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10_2024_1147_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\(C^2_+\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>C</mi> <mo>+</mo> <mn>2</mn> </msubsup> </math></EquationSource> </InlineEquation> smooth boundaries, and we prove upper bounds on the variance of the number of vertices and missed area of random <i>L</i>-polygons assuming different curvature conditions. We also transfer some of our result to a circumscribed variant of this model.</p>

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Note on the variance of generalized random polygons

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年1月10日

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  • Ferenc Fodor,
  • Balázs Grünfelder

摘要

We consider a probability model in which the hull of a sample of i.i.d. uniform random points from a convex disc K is formed by the intersection of all translates of another suitable fixed convex disc L that contain the sample. Such an object is called a random L-polygon in K. We assume that both K and L have \(C^2_+\) C + 2 smooth boundaries, and we prove upper bounds on the variance of the number of vertices and missed area of random L-polygons assuming different curvature conditions. We also transfer some of our result to a circumscribed variant of this model.