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<p>Hadwiger’s conjecture in combinatorial geometry states that any <i>n</i>-dimensional convex body can be covered by at most <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13_2025_2170_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(2^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> smaller bodies homothetic to the original body. We prove Hadwiger’s conjecture for strongly monotypic polytopes by studying a characterization of the set of normals. One of the nice properties of (strongly) monotypic polytopes is that the set of normals decides the combinatorics of the polytope.</p>

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Hadwiger’s conjecture holds for strongly monotypic polytopes

  • 期刊论文

  • 发表日期：

    2025年9月15日

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  • Vuong Bui

摘要

Hadwiger’s conjecture in combinatorial geometry states that any n-dimensional convex body can be covered by at most \(2^n\) 2 n smaller bodies homothetic to the original body. We prove Hadwiger’s conjecture for strongly monotypic polytopes by studying a characterization of the set of normals. One of the nice properties of (strongly) monotypic polytopes is that the set of normals decides the combinatorics of the polytope.