On $k-$WUR and its generalizations

TL;DR

This paper introduces generalized notions of weak uniform rotundity in Banach spaces, explores their properties, characterizations, inheritance, and applies them to infinite product spaces, extending classical geometric concepts.

Contribution

It defines $k$-WUR and $k$-WLUR, generalizes existing concepts, and characterizes these spaces via new approximation notions, also studying their inheritance and behavior in product spaces.

Findings

$k$-WUR and $k$-WLUR are characterized by $k$-weakly uniformly strong Chebyshevness.

Inheritance of $k$-WUR and $k$-WLUR by quotient spaces is established.

For infinite $ ext{ell}_p$-product spaces, $k$-WUR and $k$-WLUR are characterized, showing WUR and $k$-WUR coincide in such products.

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.