{"title":"广义二元闵科夫斯基问题的解的存在性","authors":"Mingyang Li, YanNan Liu, Jian Lu","doi":"10.1007/s12220-024-01754-y","DOIUrl":null,"url":null,"abstract":"<p>Given a real number <i>q</i> and a star body in the <i>n</i>-dimensional Euclidean space, the generalized dual curvature measure of a convex body was introduced by Lutwak et al. (Adv Math 329:85–132, 2018). The corresponding generalized dual Minkowski problem is studied in this paper. By using variational methods, we solve the generalized dual Minkowski problem for <span>\\(q<0\\)</span>, and the even generalized dual Minkowski problem for <span>\\(0\\le q\\le 1\\)</span>. We also obtain a sufficient condition for the existence of solutions to the even generalized dual Minkowski problem for <span>\\(1<q<n\\)</span>.</p>","PeriodicalId":501200,"journal":{"name":"The Journal of Geometric Analysis","volume":"171 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Existence of Solutions to the Generalized Dual Minkowski Problem\",\"authors\":\"Mingyang Li, YanNan Liu, Jian Lu\",\"doi\":\"10.1007/s12220-024-01754-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Given a real number <i>q</i> and a star body in the <i>n</i>-dimensional Euclidean space, the generalized dual curvature measure of a convex body was introduced by Lutwak et al. (Adv Math 329:85–132, 2018). The corresponding generalized dual Minkowski problem is studied in this paper. By using variational methods, we solve the generalized dual Minkowski problem for <span>\\\\(q<0\\\\)</span>, and the even generalized dual Minkowski problem for <span>\\\\(0\\\\le q\\\\le 1\\\\)</span>. We also obtain a sufficient condition for the existence of solutions to the even generalized dual Minkowski problem for <span>\\\\(1<q<n\\\\)</span>.</p>\",\"PeriodicalId\":501200,\"journal\":{\"name\":\"The Journal of Geometric Analysis\",\"volume\":\"171 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"The Journal of Geometric Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s12220-024-01754-y\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Journal of Geometric Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s12220-024-01754-y","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
给定实数q和n维欧几里得空间中的星体，Lutwak等人提出了凸体的广义对偶曲率度量（Adv Math 329:85-132, 2018）。本文研究了相应的广义对偶闵科夫斯基问题。通过使用变分法，我们求解了\(q<0\)的广义对偶Minkowski问题，以及\(0\le q\le 1\)的偶数广义对偶Minkowski问题。我们还得到了求解\(1<q<n\) 的偶数广义对偶 Minkowski 问题的充分条件。
Existence of Solutions to the Generalized Dual Minkowski Problem
Given a real number q and a star body in the n-dimensional Euclidean space, the generalized dual curvature measure of a convex body was introduced by Lutwak et al. (Adv Math 329:85–132, 2018). The corresponding generalized dual Minkowski problem is studied in this paper. By using variational methods, we solve the generalized dual Minkowski problem for \(q<0\), and the even generalized dual Minkowski problem for \(0\le q\le 1\). We also obtain a sufficient condition for the existence of solutions to the even generalized dual Minkowski problem for \(1<q<n\).
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