发表机构
Texas A&M University（德州农工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明Bourgain--Rosenthal空间的Kadets--Werner修改是渐近中点一致凸的，但因其具有Daugavet性质而缺乏连续点性质，从而否定了渐近中点一致凸与渐近一致凸的同构等价性及蕴含连续点性质的两个问题。
AI 中文摘要
设$X_{\mathsf{KW}}$是Bourgain和Rosenthal于1980年构造的$L_1$的闭子空间的Kadets--Werner修改。本文简要证明$X_{\mathsf{KW}}$是渐近中点一致凸的。由于$X_{\mathsf{KW}}$具有Daugavet性质，它缺乏连续点性质，因此不承认任何等价范数使其渐近一致凸。因此，$X_{\mathsf{KW}}$的已知性质与这一新的几何观察否定了Dilworth、Kutzarova、Randrianarivony、Revalski和Zhivkov关于渐近中点一致凸性与渐近一致凸性是否同构等价的问题，以及Perreau关于渐近中点一致凸性是否蕴含连续点性质的问题。
英文摘要
Let $X_{\mathsf{KW}}$ be the Kadets--Werner modification of a closed subspace of $L_1$ constructed by Bourgain and Rosenthal in 1980. In this short note, it is shown that $X_{\mathsf{KW}}$ is asymptotically midpoint uniformly convex. Since $X_{\mathsf{KW}}$ has the Daugavet Property, it fails the Point of Continuity Property and thus does not admit an equivalent norm that is asymptotically uniformly convex. Therefore, the previously known properties of $X_{\mathsf{KW}}$ and the new geometric observation answer, in the negative, the question of Dilworth, Kutzarova, Randrianarivony, Revalski and Zhivkov whether asymptotic midpoint uniform convexity and asymptotic uniform convexity are isomorphically equivalent and a question of Perreau whether asymptotic midpoint uniform convexity implies the Point of Continuity Property.
