论文标题
紧凑的Hölder缩回和最近的点图
Compact Hölder retractions and nearest point maps
论文作者
论文摘要
在本文中，证明了有关均匀连续缩回的两个主要结果。首先，$α$-Hölder从任何可分开的Banach空间回收到紧凑型凸子集的封闭线性跨度是整个空间，该空间都是为每个正$α<1 $构建的。这构成了戈德弗罗伊（Godefroy）和奥泽瓦（Ozawa）提出的问题的Hölder版本的积极解决方案。实际上，在某些平坦度的假设下，发现紧凑型凸组是绝对$α$-Hölder缩回。其次，我们提供了一个严格凸出的Banach Space $ x $任意接近$ \ ell_2 $（对于Banach Mazur距离）和有限的尺寸紧凑型凸子集的示例，即使最近的点映射即使限制在有限集时，最近的点映射也不均匀地连续。
In this paper, two main results concerning uniformly continuous retractions are proved. First, an $α$-Hölder retraction from any separable Banach space onto a compact convex subset whose closed linear span is the whole space is constructed for every positive $α<1$. This constitutes a positive solution to a Hölder version of a question raised by Godefroy and Ozawa. In fact, compact convex sets are found to be absolute $α$-Hölder retracts under certain assumption of flatness. Second, we provide an example of a strictly convex Banach space $X$ arbitrarily close to $\ell_2$ (for the Banach Mazur distance) and a finite dimensional compact convex subset of $X$ for which the nearest point map is not uniformly continuous even when restricted to bounded sets.
