{"title":"$L_p$积分曲率的$L_p$-Alekandrov问题","authors":"Yong Huang, E. Lutwak, Deane Yang, Gaoyong Zhang","doi":"10.4310/JDG/1536285625","DOIUrl":null,"url":null,"abstract":"It is shown that within the Lp-Brunn–Minkowski theory that Aleksandrov’s integral curvature has a natural Lp extension, for all real p. This raises the question of finding necessary and sufficient conditions on a given measure in order for it to be the Lp-integral curvature of a convex body. This problem is solved for positive p and is answered for negative p provided the given measure is even.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4310/JDG/1536285625","citationCount":"47","resultStr":"{\"title\":\"The $L_p$-Aleksandrov problem for $L_p$-integral curvature\",\"authors\":\"Yong Huang, E. Lutwak, Deane Yang, Gaoyong Zhang\",\"doi\":\"10.4310/JDG/1536285625\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"It is shown that within the Lp-Brunn–Minkowski theory that Aleksandrov’s integral curvature has a natural Lp extension, for all real p. This raises the question of finding necessary and sufficient conditions on a given measure in order for it to be the Lp-integral curvature of a convex body. This problem is solved for positive p and is answered for negative p provided the given measure is even.\",\"PeriodicalId\":15642,\"journal\":{\"name\":\"Journal of Differential Geometry\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2018-09-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.4310/JDG/1536285625\",\"citationCount\":\"47\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Differential Geometry\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4310/JDG/1536285625\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Differential Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/JDG/1536285625","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
The $L_p$-Aleksandrov problem for $L_p$-integral curvature
It is shown that within the Lp-Brunn–Minkowski theory that Aleksandrov’s integral curvature has a natural Lp extension, for all real p. This raises the question of finding necessary and sufficient conditions on a given measure in order for it to be the Lp-integral curvature of a convex body. This problem is solved for positive p and is answered for negative p provided the given measure is even.
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