Nonrough norms in spaces with small diameter

TL;DR

This paper investigates non rough norms in Banach spaces with small diameter properties, introducing new notions and establishing duality and three space properties related to these norms.

Contribution

It introduces weakly average and average non rough norms, and proves their duality and three space properties in the context of small diameter Banach spaces.

Findings

Duality between non rough norms and small diameter properties

Introduction of weakly average and average non rough norms

Proof that these norms are three space properties under certain conditions

Abstract

In this work, we study the non rough norms in Banach spaces with small diameter properties, namely the Ball Dentable Property (BDP), the Ball Huskable Property (BHP) and the Ball Small Combination of Slice Property (BSCSP). We introduce two more notions of non rough norms, namely the weakly average non rough norms and average non rough norms in Banach spaces. We prove the duality between these three versions of non rough norms and the small diameter properties in Banach spaces. We also prove that each of the three non rough norms is a three space property under certain assumptions.