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

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.