{"title":"关于 Lipschitz 近似性常数","authors":"Rubén Medina","doi":"10.1090/tran/9110","DOIUrl":null,"url":null,"abstract":"<p>In this note we find <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda greater-than 1\"> <mml:semantics> <mml:mrow> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\lambda >1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and give an explicit construction of a separable Banach space <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=\"application/x-tex\">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that there is no <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda\"> <mml:semantics> <mml:mi>λ<!-- λ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">\\lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Lipschitz retraction from <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=\"application/x-tex\">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> onto any compact convex subset of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=\"application/x-tex\">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose closed linear span is <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=\"application/x-tex\">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This is closely related to a well-known open problem raised by Godefroy and Ozawa in 2014 and represents the first known example of a Banach space with such a property.</p>","PeriodicalId":23209,"journal":{"name":"Transactions of the American Mathematical Society","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the constant of Lipschitz approximability\",\"authors\":\"Rubén Medina\",\"doi\":\"10.1090/tran/9110\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this note we find <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"lamda greater-than 1\\\"> <mml:semantics> <mml:mrow> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\lambda >1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and give an explicit construction of a separable Banach space <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper X\\\"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that there is no <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"lamda\\\"> <mml:semantics> <mml:mi>λ<!-- λ --></mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Lipschitz retraction from <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper X\\\"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> onto any compact convex subset of <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper X\\\"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose closed linear span is <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper X\\\"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This is closely related to a well-known open problem raised by Godefroy and Ozawa in 2014 and represents the first known example of a Banach space with such a property.</p>\",\"PeriodicalId\":23209,\"journal\":{\"name\":\"Transactions of the American Mathematical Society\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-01-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Transactions of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/tran/9110\",\"RegionNum\":2,\"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":"Transactions of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/tran/9110","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
在本论文中，我们找到了 λ > 1 \lambda >1，并给出了可分离巴拿赫空间 X X 的明确构造，即不存在从 X X 到 X X 的任何紧凑凸子集的 λ \lambda -Lipschitz 回缩，而 X X 的闭线性跨度是 X X。这与 Godefroy 和 Ozawa 在 2014 年提出的一个众所周知的开放问题密切相关，是具有这种性质的巴拿赫空间的第一个已知例子。
In this note we find λ>1\lambda >1 and give an explicit construction of a separable Banach space XX such that there is no λ\lambda-Lipschitz retraction from XX onto any compact convex subset of XX whose closed linear span is XX. This is closely related to a well-known open problem raised by Godefroy and Ozawa in 2014 and represents the first known example of a Banach space with such a property.
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