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A note on smooth rotund norms which are not midpoint locally uniformly rotund

Carlo Alberto De Bernardi, Alessandro Preti, Jacopo Somaglia

TL;DR

The authors construct two affirmative renorming results in infinite-dimensional Banach spaces: (i) in every separable space there exists a rotund Gâteaux smooth norm not MLUR, arbitrarily close to the original; and (ii) in every infinite-dimensional space with separable dual there exists a Fréchet smooth and WUR norm not MLUR, also arbitrarily close. These constructions advance beyond Draga's approach by producing smoothness without MLUR and solve open GMZ problems, including results about the porous nature of MLUR points. They also discuss how the MLUR set can be made porous and how the new renormings interact with URED and LUR properties. Overall, the paper deepens the geometric understanding of rotundity and smoothness in renormings and provides concrete norms with prescribed local behavior.

Abstract

We prove that every separable infinite-dimensional Banach space admits a Gâteaux smooth and rotund norm which is not midpoint locally uniformly rotund. Moreover, by using a similar technique, we provide in every infinite-dimensional Banach space with separable dual a Fréchet smooth and weakly uniformly rotund norm which is not midpoint locally uniformly rotund. These two results provide a positive answer to some open problems by A. J. Guirao, V. Montesinos, and V. Zizler.

A note on smooth rotund norms which are not midpoint locally uniformly rotund

TL;DR

The authors construct two affirmative renorming results in infinite-dimensional Banach spaces: (i) in every separable space there exists a rotund Gâteaux smooth norm not MLUR, arbitrarily close to the original; and (ii) in every infinite-dimensional space with separable dual there exists a Fréchet smooth and WUR norm not MLUR, also arbitrarily close. These constructions advance beyond Draga's approach by producing smoothness without MLUR and solve open GMZ problems, including results about the porous nature of MLUR points. They also discuss how the MLUR set can be made porous and how the new renormings interact with URED and LUR properties. Overall, the paper deepens the geometric understanding of rotundity and smoothness in renormings and provides concrete norms with prescribed local behavior.

Abstract

We prove that every separable infinite-dimensional Banach space admits a Gâteaux smooth and rotund norm which is not midpoint locally uniformly rotund. Moreover, by using a similar technique, we provide in every infinite-dimensional Banach space with separable dual a Fréchet smooth and weakly uniformly rotund norm which is not midpoint locally uniformly rotund. These two results provide a positive answer to some open problems by A. J. Guirao, V. Montesinos, and V. Zizler.
Paper Structure (7 sections, 10 theorems, 51 equations)

This paper contains 7 sections, 10 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some remarks on Draga's norm
  4. A consequence of relatively compactness of almost overcomplete sequence
  5. Main results
  6. Appendix
  7. Sum of equivalent LUR norms

Key Result

Theorem 1

Let $(X,\|\cdot\|)$ be a separable Banach space [a Banach space with separable dual, respectively], then there exists a rotund Gâteaux smooth [a weak uniformly rotund and Fréchet smooth, respectively] equivalent norm which is not midpoint locally uniformly rotund. Moreover, such a norm can be chosen

Theorems & Definitions (26)

  • Theorem 1
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • ...and 16 more