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<p>We consider the family of constant width bodies in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="454_2025_779_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {R}^3\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> which is convex under Minkowski addition. Extreme shapes cannot be expressed as a nontrivial convex combination of other constant width bodies. We show that each Meissner polyhedron is extreme. We also explain that each constant width body obtained by rotating a symmetric Reuleaux polygon about its line of symmetry is extreme. In addition, we conjecture a general characterization of all extreme constant width shapes.</p>

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On Extreme Constant Width Bodies in \(\mathbb {R}^3\)

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月27日

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  • Ryan Hynd

摘要

We consider the family of constant width bodies in \(\mathbb {R}^3\) R 3 which is convex under Minkowski addition. Extreme shapes cannot be expressed as a nontrivial convex combination of other constant width bodies. We show that each Meissner polyhedron is extreme. We also explain that each constant width body obtained by rotating a symmetric Reuleaux polygon about its line of symmetry is extreme. In addition, we conjecture a general characterization of all extreme constant width shapes.