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

TL;DR

This paper introduces methods to construct equivalent norms in Banach spaces with tailored local geometries at infinitely many points, addressing open problems related to smoothness and extreme points in the unit sphere.

Contribution

It develops tools to produce norms with specific local geometries at countably many points and solves two open problems on smoothness and extreme points in Banach spaces.

Findings

Constructed $C^k$-smooth norms that are not uniformly Gâteaux differentiable.

Produced $C^ fty$-smooth norms in $c_0( ext{Gamma})$ with dentable balls but no extreme points.

Addressed open problems from recent Banach space theory literature.

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 G\^ateaux in any direction; and on the other hand we produce in $c_0(\Gamma)$ for any infinite $\Gamma$ a $C^\infty$-smooth norm whose ball is dentable but whose sphere lacks any extreme points.