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(Asymptotic) uniform smoothness, ball separation and residuality results

Pradipta Bandyopadhyay, Deepak Gothwal

TL;DR

This work develops a ball-separation framework to study AUS and related geometric renorming properties in Banach spaces. It shows that AUS is equivalent to a ball-separation property AHUMIP, and uses this link to prove residuality of AUS, uniformly smooth, and UMIP renormings, extending known results beyond reflexive spaces. The study also leverages a duality between AUS and AUC via AUD and w*-denting points to unify several renorming phenomena. Together, these contributions provide new tools for understanding how geometric properties persist under equivalent renormings and highlight open problems on Fréchet smoothness residuality and UMIP-related super-reflexivity.

Abstract

In this article, we discuss a ball separation characterisation of asymptotically uniformly smooth (AUS) norms. We use this characterisation to prove the residuality of the set of equivalent AUS norms. We discuss similar residuality results for uniformly smooth norms and norms with uniform Mazur intersection property (UMIP).

(Asymptotic) uniform smoothness, ball separation and residuality results

TL;DR

This work develops a ball-separation framework to study AUS and related geometric renorming properties in Banach spaces. It shows that AUS is equivalent to a ball-separation property AHUMIP, and uses this link to prove residuality of AUS, uniformly smooth, and UMIP renormings, extending known results beyond reflexive spaces. The study also leverages a duality between AUS and AUC via AUD and w*-denting points to unify several renorming phenomena. Together, these contributions provide new tools for understanding how geometric properties persist under equivalent renormings and highlight open problems on Fréchet smoothness residuality and UMIP-related super-reflexivity.

Abstract

In this article, we discuss a ball separation characterisation of asymptotically uniformly smooth (AUS) norms. We use this characterisation to prove the residuality of the set of equivalent AUS norms. We discuss similar residuality results for uniformly smooth norms and norms with uniform Mazur intersection property (UMIP).

Paper Structure

This paper contains 11 sections, 24 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. AUD characterises AUC
  4. Ball separation characterisation of AUS
  5. Residuality Results
  6. Residuality of AUS norms
  7. Residuality of uniformly smooth norms
  8. Residuality of norms with UMIP
  9. Two open problems
  10. Residuality of Fréchet smooth norms
  11. Uniformly convex renorming and UMIP

Key Result

Theorem 3.1

$X$ is AUS if and only if $X^*$ is w*-AUC.

Theorems & Definitions (41)

  • Definition 2.1
  • Theorem 3.1
  • Definition 3.2
  • Lemma 3.3
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • Definition 4.1
  • Definition 4.2
  • Lemma 4.3
  • ...and 31 more