For every convex body <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2143_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$K \subset \mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2143_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta \in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, the <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2143_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>-convolution body of K is the set of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2143_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$x \in \mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> for which <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2143_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="181" /> </InlineMediaObject> <EquationSource Format="TEX">${\left| K \cap (K+x)\right| }_{n}\ge \delta {\left| K\right| }_{n}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mfenced close="|" open="|"> <mi>K</mi> <mo>∩</mo> <mo stretchy="false">(</mo> <mi>K</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mfenced> <mi>n</mi> </msub> <mo>≥</mo> <mi>δ</mi> <msub> <mfenced close="|" open="|"> <mi>K</mi> </mfenced> <mi>n</mi> </msub> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We show that for <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2143_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$n=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and any <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2143_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta \in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, ellipsoids do not maximize the volume of the <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2143_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>-convolution body of K, when K runs over all convex bodies of a fixed volume. This behavior is somehow unexpected and contradicts the limit case <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2143_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta \rightarrow 1^-$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo stretchy="false">→</mo> <msup> <mn>1</mn> <mo>-</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, which is governed by the Petty projection inequality.

On the volume of convolution bodies in the plane

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2025年8月18日

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J. Haddad

摘要

For every convex body $K \subset \mathbb {R}^n$ $K \subset R^{n}$ and $\delta \in (0,1)$ $δ \in (0, 1)$ , the $\delta $ $δ$ -convolution body of K is the set of $x \in \mathbb {R}^n$ $x \in R^{n}$ for which ${\left| K \cap (K+x)\right| }_{n}\ge \delta {\left| K\right| }_{n}.$ ${|K \cap (K + x)|}_{n} \geq δ {|K|}_{n} .$ We show that for $n=2$ $n = 2$ and any $\delta \in (0,1)$ $δ \in (0, 1)$ , ellipsoids do not maximize the volume of the $\delta $ $δ$ -convolution body of K, when K runs over all convex bodies of a fixed volume. This behavior is somehow unexpected and contradicts the limit case $\delta \rightarrow 1^-$ $δ \to 1^{-}$ , which is governed by the Petty projection inequality.