论文标题
在Banach空间和应用领域的许多点上保存本地几何形状的重新分类
Renormings preserving local geometry at countably many points in spheres of Banach spaces and applications
论文作者
论文摘要
我们开发了通过电感程序在Banach空间球体中无限多个点围绕特定局部几何形状生成等效规范的工具。我们将这个过程与平滑度结果和技术相结合，以解决A. J. Guirao，V。Montesinos和V. Zizler的最近发表的专着[GMZ22]中提出的两个开放问题。具体而言，一方面，我们在每个可分离的Banach空间中构建了$ C^k $ - 平滑规范的等效标准，即$ C^k $ - smoth，但在任何方向上都无法统一地gâteaux；另一方面，我们以任何无限的$γ$ a $ c^\ infty $ - s-smooth norm的范围生产$ c_0（γ）$，其球是可凹的，但其球体缺乏任何极端点。
We develop tools to produce equivalent norms with specific local geometry around infinitely many points in the sphere of a Banach space via an inductive procedure. We combine this process with smoothness results and techniques to solve two open problems posed in the recently published monograph [GMZ22] by A. J. Guirao, V. Montesinos and V. Zizler. Specifically, on the one hand we construct in every separable Banach space admitting a $C^k$-smooth norm an equivalent norm which is $C^k$-smooth but fails to be uniformly Gâteaux in any direction; and on the other hand we produce in $c_0(Γ)$ for any infinite $Γ$ a $C^\infty$-smooth norm whose ball is dentable but whose sphere lacks any extreme points.
