Rotund Gateaux smooth norms which are not locally uniformly rotund

TL;DR

This paper constructs a specific type of smooth norm in infinite-dimensional Banach spaces that is rotund but not locally uniformly rotund, solving an open problem in the field.

Contribution

It demonstrates the existence of rotund Gateaux smooth norms that are not locally uniformly rotund in every infinite-dimensional separable Banach space, addressing a previously open question.

Findings

Existence of such norms in all infinite-dimensional separable Banach spaces

Construction of an average locally uniformly rotund (LUR) Gateaux smooth norm

Resolution of an open problem by Guirao, Montesinos, and Zizler

Abstract

We provide, in every infinite-dimensional separable Banach space, an average locally uniformly rotund (and hence rotund) Gateaux smooth renorming which is not locally uniformly rotund. This solves an open problem posed by A.J. Guirao, V. Montesinos, and V. Zizler.