{"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}
引用次数: 0
Abstract
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.
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。
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