Some remarks on $p$-median problem

TL;DR

This paper explores the properties of the p-median problem within Banach spaces, establishing conditions under which certain subspaces are proximinal in Lp spaces, thus advancing the theoretical understanding of approximation in functional analysis.

Contribution

It provides new theoretical results linking proximinality in Banach spaces to properties of subspaces in Lp spaces, specifically for weakly K-analytic subspaces.

Findings

L_p(μ,Y) is p-simultaneously proximinal in L_p(μ,X) under certain conditions

Proximinality in Y implies proximinality in L_p spaces

Advances understanding of approximation properties in Banach space theory

Abstract

It is shown that for a given Banach space $X$ and a subspace $Y$ weakly $\mathcal{K}$-analytic, $L_p(\mu,Y)$ is $p$-simultaneously proximinal in $L_p(\mu,X)$ whenever $Y$ is $p$-simultaneously proximinal in $X$.