{"title":"Affirmative Resolution of Bourgain’s Slicing Problem Using Guan’s Bound","authors":"Boaz Klartag, Joseph Lehec","doi":"10.1007/s00039-025-00718-w","DOIUrl":null,"url":null,"abstract":"<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 <span><span style=\"\">K \\subseteq \\mathbb{R}^{n}</span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"2.313ex\" role=\"img\" style=\"vertical-align: -0.505ex;\" viewbox=\"0 -778.3 3470.7 995.9\" width=\"8.061ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-4B\" y=\"0\"></use><use x=\"1167\" xlink:href=\"#MJMAIN-2286\" y=\"0\"></use><g transform=\"translate(2223,0)\"><use x=\"0\" xlink:href=\"#MJAMS-52\" y=\"0\"></use><use transform=\"scale(0.707)\" x=\"1021\" xlink:href=\"#MJMATHI-6E\" y=\"581\"></use></g></g></svg></span><script type=\"math/tex\">K \\subseteq \\mathbb{R}^{n}</script></span> of volume one, there exists a hyperplane <span><span style=\"\">H \\subseteq \\mathbb{R}^{n}</span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"2.309ex\" role=\"img\" style=\"vertical-align: -0.505ex;\" viewbox=\"0 -777 3469.7 994.3\" width=\"8.059ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-48\" y=\"0\"></use><use x=\"1166\" xlink:href=\"#MJMAIN-2286\" y=\"0\"></use><g transform=\"translate(2222,0)\"><use x=\"0\" xlink:href=\"#MJAMS-52\" y=\"0\"></use><use transform=\"scale(0.707)\" x=\"1021\" xlink:href=\"#MJMATHI-6E\" y=\"581\"></use></g></g></svg></span><script type=\"math/tex\">H \\subseteq \\mathbb{R}^{n}</script></span> such that </p><span><span style=\"\"> Vol_{n-1}(K \\cap H) > c, </span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"2.614ex\" role=\"img\" style=\"vertical-align: -0.706ex;\" viewbox=\"0 -821.4 8697.5 1125.3\" width=\"20.201ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-56\" y=\"0\"></use><use x=\"769\" xlink:href=\"#MJMATHI-6F\" y=\"0\"></use><g transform=\"translate(1255,0)\"><use x=\"0\" xlink:href=\"#MJMATHI-6C\" y=\"0\"></use><g transform=\"translate(298,-150)\"><use transform=\"scale(0.707)\" x=\"0\" xlink:href=\"#MJMATHI-6E\" y=\"0\"></use><use transform=\"scale(0.707)\" x=\"600\" xlink:href=\"#MJMAIN-2212\" y=\"0\"></use><use transform=\"scale(0.707)\" x=\"1379\" xlink:href=\"#MJMAIN-31\" y=\"0\"></use></g></g><use x=\"2982\" xlink:href=\"#MJMAIN-28\" y=\"0\"></use><use x=\"3372\" xlink:href=\"#MJMATHI-4B\" y=\"0\"></use><use x=\"4483\" xlink:href=\"#MJMAIN-2229\" y=\"0\"></use><use x=\"5373\" xlink:href=\"#MJMATHI-48\" y=\"0\"></use><use x=\"6261\" xlink:href=\"#MJMAIN-29\" y=\"0\"></use><use x=\"6929\" xlink:href=\"#MJMAIN-3E\" y=\"0\"></use><use x=\"7985\" xlink:href=\"#MJMATHI-63\" y=\"0\"></use><use x=\"8419\" xlink:href=\"#MJMAIN-2C\" y=\"0\"></use></g></svg></span><script type=\"math/tex; mode=display\"> Vol_{n-1}(K \\cap H) > c, </script></span><p> where <span><span style=\"\">c > 0</span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"1.912ex\" role=\"img\" style=\"vertical-align: -0.205ex;\" viewbox=\"0 -735.2 2268.1 823.4\" width=\"5.268ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-63\" y=\"0\"></use><use x=\"711\" xlink:href=\"#MJMAIN-3E\" y=\"0\"></use><use x=\"1767\" xlink:href=\"#MJMAIN-30\" y=\"0\"></use></g></svg></span><script type=\"math/tex\">c > 0</script></span> is a universal constant. Our proof combines Milman’s theory of <span><span style=\"\">M</span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"1.912ex\" role=\"img\" style=\"vertical-align: -0.205ex;\" viewbox=\"0 -735.2 1051.5 823.4\" width=\"2.442ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-4D\" y=\"0\"></use></g></svg></span><script type=\"math/tex\">M</script></span>-ellipsoids, stochastic localization with a recent bound by Guan, and stability estimates for the Shannon-Stam inequality by Eldan and Mikulincer.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"28 1","pages":""},"PeriodicalIF":2.5000,"publicationDate":"2025-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Geometric and Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00039-025-00718-w","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
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 \mathbb{R}^{n} of volume one, there exists a hyperplane H \subseteq \mathbb{R}^{n} such that
Vol_{n-1}(K \cap H) > c,
where c > 0 is a universal constant. Our proof combines Milman’s theory of M-ellipsoids, stochastic localization with a recent bound by Guan, and stability estimates for the Shannon-Stam inequality by Eldan and Mikulincer.
期刊介绍:
Geometric And Functional Analysis (GAFA) publishes original research papers of the highest quality on a broad range of mathematical topics related to geometry and analysis.
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Publishes major results on topics in geometry and analysis.
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