{"title":"凸包络的可微性","authors":"Bernd Kirchheim , Jan Kristensen","doi":"10.1016/S0764-4442(01)02117-6","DOIUrl":null,"url":null,"abstract":"<div><p>We prove that the convex envelope of a differentiable, or C<sup>1,<em>α</em></sup>-function <em>f</em> is C<sup>1</sup>, or C<sup>1,<em>α</em></sup> respectively, provided only that the function satisfies the very mild growth condition that <em>f</em>(<em>x</em>) tends to +∞ if |<em>x</em>| does so.</p></div>","PeriodicalId":100300,"journal":{"name":"Comptes Rendus de l'Académie des Sciences - Series I - Mathematics","volume":"333 8","pages":"Pages 725-728"},"PeriodicalIF":0.0000,"publicationDate":"2001-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0764-4442(01)02117-6","citationCount":"49","resultStr":"{\"title\":\"Differentiability of convex envelopes\",\"authors\":\"Bernd Kirchheim , Jan Kristensen\",\"doi\":\"10.1016/S0764-4442(01)02117-6\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We prove that the convex envelope of a differentiable, or C<sup>1,<em>α</em></sup>-function <em>f</em> is C<sup>1</sup>, or C<sup>1,<em>α</em></sup> respectively, provided only that the function satisfies the very mild growth condition that <em>f</em>(<em>x</em>) tends to +∞ if |<em>x</em>| does so.</p></div>\",\"PeriodicalId\":100300,\"journal\":{\"name\":\"Comptes Rendus de l'Académie des Sciences - Series I - Mathematics\",\"volume\":\"333 8\",\"pages\":\"Pages 725-728\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2001-10-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/S0764-4442(01)02117-6\",\"citationCount\":\"49\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Comptes Rendus de l'Académie des Sciences - Series I - Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0764444201021176\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Comptes Rendus de l'Académie des Sciences - Series I - Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0764444201021176","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We prove that the convex envelope of a differentiable, or C1,α-function f is C1, or C1,α respectively, provided only that the function satisfies the very mild growth condition that f(x) tends to +∞ if |x| does so.
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