Diameter Two Properties, Convexity and Smoothness

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• 作者：Trond A. Abrahamsen ; Vegard Lima ; Olav Nygaard…
• 关键词：Diameter 2 property ; strictly convex norm ; smooth norm ; Daugavet property
• 刊名：Milan Journal of Mathematics
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：84
• 期：2
• 页码：231-242
• 全文大小：711 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1424-9294
• 卷排序：84

文摘

We study smoothness and strict convexity of (the bidual) of Banach spaces in the presence of diameter 2 properties.We prove that the strong diameter 2 property prevents the bidual from being strictly convex and being smooth, and we initiate the investigation whether the same is true for the (local) diameter 2 property. We also give characterizations of the following property for a Banach space \({X}\): “For every slice \({S}\) of \({B_X}\) and every norm-one element \({x}\) in \({S}\), there is a point \({y \in S}\) in distance as close to 2 as we want.” Spaces with this property are shown to have non-smooth bidual.