{"title":"平面𝐿_{𝑝} 对偶闵科夫斯基问题的直径估计","authors":"Minhyun Kim, Taehun Lee","doi":"10.1090/proc/16464","DOIUrl":null,"url":null,"abstract":"<p>In this paper, given a prescribed measure on <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper S Superscript 1\">\n <mml:semantics>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">S</mml:mi>\n </mml:mrow>\n <mml:mn>1</mml:mn>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {S}^1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> whose density is bounded and positive, we establish a uniform diameter estimate for solutions to the planar <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript p\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mi>p</mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">L_p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> dual Minkowski problem when <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0 greater-than p greater-than 1\">\n <mml:semantics>\n <mml:mrow>\n <mml:mn>0</mml:mn>\n <mml:mo>></mml:mo>\n <mml:mi>p</mml:mi>\n <mml:mo>></mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">0>p>1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"q greater-than-or-equal-to 2\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>q</mml:mi>\n <mml:mo>≥</mml:mo>\n <mml:mn>2</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">q\\ge 2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. We also prove the uniqueness and positivity of solutions to the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript p\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>L</mml:mi>\n <mml:mi>p</mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">L_p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> Minkowski problem when the density of the measure is sufficiently close to a constant in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript alpha\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>C</mml:mi>\n <mml:mi>α</mml:mi>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">C^\\alpha</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>.</p>","PeriodicalId":0,"journal":{"name":"","volume":"53 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Diameter estimate for planar 𝐿_{𝑝} dual Minkowski problem\",\"authors\":\"Minhyun Kim, Taehun Lee\",\"doi\":\"10.1090/proc/16464\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper, given a prescribed measure on <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper S Superscript 1\\\">\\n <mml:semantics>\\n <mml:msup>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">S</mml:mi>\\n </mml:mrow>\\n <mml:mn>1</mml:mn>\\n </mml:msup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {S}^1</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> whose density is bounded and positive, we establish a uniform diameter estimate for solutions to the planar <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L Subscript p\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mi>L</mml:mi>\\n <mml:mi>p</mml:mi>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">L_p</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> dual Minkowski problem when <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"0 greater-than p greater-than 1\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mn>0</mml:mn>\\n <mml:mo>></mml:mo>\\n <mml:mi>p</mml:mi>\\n <mml:mo>></mml:mo>\\n <mml:mn>1</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">0>p>1</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"q greater-than-or-equal-to 2\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>q</mml:mi>\\n <mml:mo>≥</mml:mo>\\n <mml:mn>2</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">q\\\\ge 2</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. We also prove the uniqueness and positivity of solutions to the <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L Subscript p\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mi>L</mml:mi>\\n <mml:mi>p</mml:mi>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">L_p</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> Minkowski problem when the density of the measure is sufficiently close to a constant in <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper C Superscript alpha\\\">\\n <mml:semantics>\\n <mml:msup>\\n <mml:mi>C</mml:mi>\\n <mml:mi>α</mml:mi>\\n </mml:msup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">C^\\\\alpha</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":\"53 3\",\"pages\":\"\"},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-05-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/proc/16464\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16464","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
在本文中，给定 S 1 \mathbb {S}^1 上密度有界且为正的规定度量，当 0 > p > 1 0>p>1 且 q≥ 2 q\ge 2 时，我们建立了平面 L p L_p 对偶闵科夫斯基问题解的均匀直径估计。我们还证明了当度量密度足够接近 C α C^\alpha 中的一个常数时，L p L_p Minkowski 问题解的唯一性和实在性。
Diameter estimate for planar 𝐿_{𝑝} dual Minkowski problem
In this paper, given a prescribed measure on S1\mathbb {S}^1 whose density is bounded and positive, we establish a uniform diameter estimate for solutions to the planar LpL_p dual Minkowski problem when 0>p>10>p>1 and q≥2q\ge 2. We also prove the uniqueness and positivity of solutions to the LpL_p Minkowski problem when the density of the measure is sufficiently close to a constant in CαC^\alpha.
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