<p>We provide the final step in the resolution of Bourgain’s slicing problem in the affirmative. Thus we establish the following theorem: for any convex body <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="39_2025_718_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>K</mi> <mo>⊆</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>n</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$K \subseteq \mathbb{R}^{n}$</EquationSource> </InlineEquation> of volume one, there exists a hyperplane <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="39_2025_718_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>H</mi> <mo>⊆</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>n</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$H \subseteq \mathbb{R}^{n}$</EquationSource> </InlineEquation> such that <Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="39_2025_718_Article_Equa.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="148" /> </MediaObject> <EquationSource Format="MATHML"><math> <mi>V</mi> <mi>o</mi> <msub> <mi>l</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>K</mi> <mo>∩</mo> <mi>H</mi> <mo stretchy="false">)</mo> <mo>></mo> <mi>c</mi> <mo>,</mo> </math></EquationSource> <EquationSource Format="TEX">$ Vol_{n-1}(K \cap H) > c, $</EquationSource> </Equation> where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="39_2025_718_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>c</mi> <mo>></mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$c > 0$</EquationSource> </InlineEquation> is a universal constant. Our proof combines Milman’s theory of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="39_2025_718_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>M</mi> </math></EquationSource> <EquationSource Format="TEX">$M$</EquationSource> </InlineEquation>-ellipsoids, stochastic localization with a recent bound by Guan, and stability estimates for the Shannon-Stam inequality by Eldan and Mikulincer.</p>

Affirmative Resolution of Bourgain’s Slicing Problem Using Guan’s Bound

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2025年8月1日

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Boaz Klartag,
Joseph Lehec

摘要

We provide the final step in the resolution of Bourgain’s slicing problem in the affirmative. Thus we establish the following theorem: for any convex body $K \subseteq R^{n}$ $K \subseteq \mathbb{R}^{n}$ of volume one, there exists a hyperplane $H \subseteq R^{n}$ $H \subseteq \mathbb{R}^{n}$ such that $V o l_{n - 1} (K \cap H) > c,$ $ Vol_{n-1}(K \cap H) > c, $ where $c > 0$ $c > 0$ is a universal constant. Our proof combines Milman’s theory of $M$ $M$ -ellipsoids, stochastic localization with a recent bound by Guan, and stability estimates for the Shannon-Stam inequality by Eldan and Mikulincer.