A lattice (d, k)-polytope is the convex hull of a set of points in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10013_2025_742_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> whose coordinates are integers ranging between 0 and k. We consider the smallest possible distance <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10013_2025_742_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon (d,k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> between two disjoint lattice (d, k)-polytopes. We propose an algebraic model for this distance and derive from it an explicit formula for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10013_2025_742_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon (2,k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Our model also allows for the computation of previously intractable values of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10013_2025_742_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon (d,k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In particular, we compute <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10013_2025_742_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon (3,k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10013_2025_742_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$4\le {k}\le 8$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>4</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mn>8</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10013_2025_742_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon (4,k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10013_2025_742_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$2\le {k}\le 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10013_2025_742_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon (6,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo stretchy="false">(</mo> <mn>6</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

Small Kissing Polytopes

期刊论文
发表日期：

2025年4月10日

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Antoine Deza,
Zhongyuan Liu,
Lionel Pournin

摘要

A lattice (d, k)-polytope is the convex hull of a set of points in $\mathbb {R}^d$ $R^{d}$ whose coordinates are integers ranging between 0 and k. We consider the smallest possible distance $\varepsilon (d,k)$ $ε (d, k)$ between two disjoint lattice (d, k)-polytopes. We propose an algebraic model for this distance and derive from it an explicit formula for $\varepsilon (2,k)$ $ε (2, k)$ . Our model also allows for the computation of previously intractable values of $\varepsilon (d,k)$ $ε (d, k)$ . In particular, we compute $\varepsilon (3,k)$ $ε (3, k)$ when $4\le {k}\le 8$ $4 \leq k \leq 8$ , $\varepsilon (4,k)$ $ε (4, k)$ when $2\le {k}\le 3$ $2 \leq k \leq 3$ , and $\varepsilon (6,1)$ $ε (6, 1)$ .