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On $k-$WUR and its generalizations

P. Gayathri, Vamsinadh Thota

TL;DR

The paper develops a comprehensive framework for $k$-generalized rotundity notions in real Banach spaces, introducing $k$-WUR and $k$-WLUR along with $k$-wSCh and $k$-wUSCh as tools to capture best-approximation behavior. It furnishes sequential and ball-based characterizations, links these properties to quotient structures, and analyzes stability under $\,ell_p$-product constructions. A major contribution is the characterization of $k$-WUR/$k$-WLUR in terms of $k$-wUSCh and the demonstration that infinite $\,ell_p$-products yield equivalences with their classical WUR/WLUR counterparts. The results unify and extend prior work on $k$-rotundity, providing exact conditions for inheritance in quotients and products and highlighting the nuanced relationships among generalized rotundity notions.

Abstract

We introduce two notions called $k-$weakly uniform rotundity ($k-$WUR) and $k-$weakly locally uniform rotundity ($k-$WLUR) in real Banach spaces. These are natural generalizations of the well-known concepts $k-$UR and WUR. By introducing two best approximation notions namely $k-$weakly strong Chebyshevity and $k-$weakly uniform strong Chebyshevity, we generalize some of the existing results to $k-$WUR and $k-$WLUR spaces. In particular, we present characterizations of $k-$WUR spaces in terms of $k-$weakly uniformly strong Chebyshevness. Also, the inheritance of the notions $k-$WUR and $k-$WLUR by quotient spaces are discussed. Further, we provide a necessary and sufficient condition for an infinite $\ell_p-$product space to be $k-$WUR (respectively, $k-$WLUR). As a consequence, we observe that the notions WUR and $k-$WUR coincide for an infinite $\ell_p-$product of a Banach space.

On $k-$WUR and its generalizations

TL;DR

The paper develops a comprehensive framework for -generalized rotundity notions in real Banach spaces, introducing -WUR and -WLUR along with -wSCh and -wUSCh as tools to capture best-approximation behavior. It furnishes sequential and ball-based characterizations, links these properties to quotient structures, and analyzes stability under -product constructions. A major contribution is the characterization of -WUR/-WLUR in terms of -wUSCh and the demonstration that infinite -products yield equivalences with their classical WUR/WLUR counterparts. The results unify and extend prior work on -rotundity, providing exact conditions for inheritance in quotients and products and highlighting the nuanced relationships among generalized rotundity notions.

Abstract

We introduce two notions called weakly uniform rotundity (WUR) and weakly locally uniform rotundity (WLUR) in real Banach spaces. These are natural generalizations of the well-known concepts UR and WUR. By introducing two best approximation notions namely weakly strong Chebyshevity and weakly uniform strong Chebyshevity, we generalize some of the existing results to WUR and WLUR spaces. In particular, we present characterizations of WUR spaces in terms of weakly uniformly strong Chebyshevness. Also, the inheritance of the notions WUR and WLUR by quotient spaces are discussed. Further, we provide a necessary and sufficient condition for an infinite product space to be WUR (respectively, WLUR). As a consequence, we observe that the notions WUR and WUR coincide for an infinite product of a Banach space.
Paper Structure (4 sections, 35 theorems, 62 equations)

This paper contains 4 sections, 35 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Characterizations of $k-$WUR, $k-$WLUR and $k-$WMLUR
  4. Stability of $k-$WUR, $k-$WLUR and $k-$WMLUR

Key Result

Lemma 2.2

Let $(x_n^{(1)}),(x_n^{(2)}), \dots, (x_n^{(k+1)})$ be $(k+1)-$bounded sequences in $X$. Then for any $f_1, f_2, \dots, f_{k} \in S_{X^*}$ the following statements hold.

Theorems & Definitions (80)

  • Definition 1.1
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Example 2.6
  • ...and 70 more