On a generalisation of Local Uniform Rotundity

TL;DR

This paper explores the (HLUR) property, a generalization of local uniform rotundity in Banach spaces, characterizing it through geometric properties and examining its implications for finite-dimensional spaces and farthest points.

Contribution

It introduces and characterizes the (HLUR) property, linking it to known geometric properties and demonstrating its equivalence with the anti-Daugavet property in finite dimensions.

Findings

(HLUR) coincides with anti-Daugavet property in finite-dimensional spaces

Characterization of (HLUR) via geometric properties

Applications to farthest points in Banach spaces

Abstract

In this paper we investigate the property (HLUR), a generalisation of (LUR) property of a Banach space. A Banach space having the property (HLUR) is called an HLUR space. We characterise (HLUR) property with the help of known geometric properties and study various properties of HLUR spaces. We show that for any finite dimensional Banach space, the property (HLUR) coincides with anti-Daugavet property of the space. We also show some applications of HLUR spaces in connection with farthest points of sets.