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<p>Bruckner (Math Mag 37:250–251, 1964) observed that any bounded open set in the plane has an inscribed triangle, that is a triangle contained in the open set and with the vertices lying on the boundary. We prove that this triangle can be taken uniformly fat, more precisely having all internal angles larger than <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="454_2025_757_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sim 0.3\)</EquationSource> </InlineEquation>° and area bounded below by a constant multiple of the area of the largest inscribed ball, independently of the choice of the initial open set. We also build a polygon in which all the inscribed triangles are not too-fat, meaning that at least one angle is less than <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="454_2025_757_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sim 55\)</EquationSource> </InlineEquation>°. These results show the existence of a maximal number <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="454_2025_757_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Theta\)</EquationSource> </InlineEquation> strictly between 0 and 60, whose exact value remains unknown, for which all bounded open sets admit an inscribed triangle with all angles larger than or equal to <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="454_2025_757_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Theta\)</EquationSource> </InlineEquation> degrees.</p>

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Fat Triangles Inscribed in Arbitrary Planar Domains

  • 期刊论文

  • 发表日期：

    2025年8月29日

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  • Ivan Yuri Violo

摘要

Bruckner (Math Mag 37:250–251, 1964) observed that any bounded open set in the plane has an inscribed triangle, that is a triangle contained in the open set and with the vertices lying on the boundary. We prove that this triangle can be taken uniformly fat, more precisely having all internal angles larger than \(\sim 0.3\) ° and area bounded below by a constant multiple of the area of the largest inscribed ball, independently of the choice of the initial open set. We also build a polygon in which all the inscribed triangles are not too-fat, meaning that at least one angle is less than \(\sim 55\) °. These results show the existence of a maximal number \(\Theta\) strictly between 0 and 60, whose exact value remains unknown, for which all bounded open sets admit an inscribed triangle with all angles larger than or equal to \(\Theta\) degrees.