{"title":"近似凸性的急剧加倍阈值","authors":"Peter van Hintum, Peter Keevash","doi":"10.1112/blms.13129","DOIUrl":null,"url":null,"abstract":"<p>We show for <span></span><math>\n <semantics>\n <mrow>\n <mi>A</mi>\n <mo>,</mo>\n <mi>B</mi>\n <mo>⊂</mo>\n <msup>\n <mi>R</mi>\n <mi>d</mi>\n </msup>\n </mrow>\n <annotation>$A,B\\subset \\mathbb {R}^d$</annotation>\n </semantics></math> of equal volume and <span></span><math>\n <semantics>\n <mrow>\n <mi>t</mi>\n <mo>∈</mo>\n <mo>(</mo>\n <mn>0</mn>\n <mo>,</mo>\n <mn>1</mn>\n <mo>/</mo>\n <mn>2</mn>\n <mo>]</mo>\n </mrow>\n <annotation>$t\\in (0,1/2]$</annotation>\n </semantics></math> that if <span></span><math>\n <semantics>\n <mrow>\n <mrow>\n <mo>|</mo>\n <mi>t</mi>\n <mi>A</mi>\n <mo>+</mo>\n <mrow>\n <mo>(</mo>\n <mn>1</mn>\n <mo>−</mo>\n <mi>t</mi>\n <mo>)</mo>\n </mrow>\n <mi>B</mi>\n <mo>|</mo>\n <mo><</mo>\n </mrow>\n <mrow>\n <mo>(</mo>\n <mn>1</mn>\n <mo>+</mo>\n <msup>\n <mi>t</mi>\n <mi>d</mi>\n </msup>\n <mo>)</mo>\n </mrow>\n <mrow>\n <mo>|</mo>\n <mi>A</mi>\n <mo>|</mo>\n </mrow>\n </mrow>\n <annotation>$|tA+(1-t)B|&lt; (1+t^d)|A|$</annotation>\n </semantics></math>, then (up to translation) <span></span><math>\n <semantics>\n <mrow>\n <mo>|</mo>\n <mo>co</mo>\n <mo>(</mo>\n <mi>A</mi>\n <mo>∪</mo>\n <mi>B</mi>\n <mo>)</mo>\n <mo>|</mo>\n <mo>/</mo>\n <mo>|</mo>\n <mi>A</mi>\n <mo>|</mo>\n </mrow>\n <annotation>$|\\operatorname{co}(A\\cup B)|/|A|$</annotation>\n </semantics></math> is bounded. This establishes the sharp threshold for the quantitative stability of the Brunn–Minkowski inequality recently established by Figalli, van Hintum, and Tiba, the proof of which uses our current result. We additionally establish a similar sharp threshold for iterated sumsets.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 10","pages":"3229-3239"},"PeriodicalIF":0.8000,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13129","citationCount":"0","resultStr":"{\"title\":\"The sharp doubling threshold for approximate convexity\",\"authors\":\"Peter van Hintum, Peter Keevash\",\"doi\":\"10.1112/blms.13129\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We show for <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>A</mi>\\n <mo>,</mo>\\n <mi>B</mi>\\n <mo>⊂</mo>\\n <msup>\\n <mi>R</mi>\\n <mi>d</mi>\\n </msup>\\n </mrow>\\n <annotation>$A,B\\\\subset \\\\mathbb {R}^d$</annotation>\\n </semantics></math> of equal volume and <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>t</mi>\\n <mo>∈</mo>\\n <mo>(</mo>\\n <mn>0</mn>\\n <mo>,</mo>\\n <mn>1</mn>\\n <mo>/</mo>\\n <mn>2</mn>\\n <mo>]</mo>\\n </mrow>\\n <annotation>$t\\\\in (0,1/2]$</annotation>\\n </semantics></math> that if <span></span><math>\\n <semantics>\\n <mrow>\\n <mrow>\\n <mo>|</mo>\\n <mi>t</mi>\\n <mi>A</mi>\\n <mo>+</mo>\\n <mrow>\\n <mo>(</mo>\\n <mn>1</mn>\\n <mo>−</mo>\\n <mi>t</mi>\\n <mo>)</mo>\\n </mrow>\\n <mi>B</mi>\\n <mo>|</mo>\\n <mo><</mo>\\n </mrow>\\n <mrow>\\n <mo>(</mo>\\n <mn>1</mn>\\n <mo>+</mo>\\n <msup>\\n <mi>t</mi>\\n <mi>d</mi>\\n </msup>\\n <mo>)</mo>\\n </mrow>\\n <mrow>\\n <mo>|</mo>\\n <mi>A</mi>\\n <mo>|</mo>\\n </mrow>\\n </mrow>\\n <annotation>$|tA+(1-t)B|&lt; (1+t^d)|A|$</annotation>\\n </semantics></math>, then (up to translation) <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>|</mo>\\n <mo>co</mo>\\n <mo>(</mo>\\n <mi>A</mi>\\n <mo>∪</mo>\\n <mi>B</mi>\\n <mo>)</mo>\\n <mo>|</mo>\\n <mo>/</mo>\\n <mo>|</mo>\\n <mi>A</mi>\\n <mo>|</mo>\\n </mrow>\\n <annotation>$|\\\\operatorname{co}(A\\\\cup B)|/|A|$</annotation>\\n </semantics></math> is bounded. This establishes the sharp threshold for the quantitative stability of the Brunn–Minkowski inequality recently established by Figalli, van Hintum, and Tiba, the proof of which uses our current result. We additionally establish a similar sharp threshold for iterated sumsets.</p>\",\"PeriodicalId\":55298,\"journal\":{\"name\":\"Bulletin of the London Mathematical Society\",\"volume\":\"56 10\",\"pages\":\"3229-3239\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-07-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13129\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the London Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/blms.13129\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/blms.13129","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我们证明，对于 A , B ⊂ R d $A,B\subset \mathbb {R}^d$ 体积相等且 t∈ ( 0 , 1 / 2 ]$，如果 | t A + ( 1 - t ) 在 (0,1/2]$ 中，则 | t A + ( 1 - t ) = $。 $t\in (0,1/2]$ that if | t A + ( 1 - t ) B | < ( 1 + t d ) | A | $|tA+(1-t)B|< (1+t^d)|A|$ ，则（直至平移） | co ( A ∪ B ) | / | A | $|\operatorname{co}(A\cup B)|/|A|$ 是有界的。这就确立了菲加里、范欣图姆和蒂巴最近建立的布鲁恩-明考斯基不等式定量稳定性的尖锐阈值，其证明使用了我们当前的结果。我们还为迭代和集建立了类似的尖锐临界值。
The sharp doubling threshold for approximate convexity
We show for of equal volume and that if , then (up to translation) is bounded. This establishes the sharp threshold for the quantitative stability of the Brunn–Minkowski inequality recently established by Figalli, van Hintum, and Tiba, the proof of which uses our current result. We additionally establish a similar sharp threshold for iterated sumsets.
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
