Renormings preserving local geometry at countably many points in spheres of Banach spaces and applications

2022/11/22 by Andrés Quilis, Quilis, Andrés
Computer Science · Mathematics · #46B03 #46B10 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.2211.12332

openalex publication_date 2022/11/22 · openalex created_date 2022/11/29 · openalex updated_date 2026/07/28

Abstract

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 c₀(Γ) for any infinite Γ a C^∞-smooth norm whose ball is dentable but whose sphere lacks any extreme points.

Related

• A note on smooth rotund norms which are not midpoint locally uniformly rotund
• An asymptotic analog of a local-to-global phenomenon for uniformly convex renormings
• Some Classical and Recent Results Concerning Renorming Theory
• Three-space property for asymptotically uniformly smooth renormings
• Some remarks on almost locally uniformly rotund points