Best \(N\)-simultaneous approximation in \(L_p(\mu, X)\) (Q2405882)

Summary: Let \(X\) be a Banach space. Let \(1 \leq p < \infty\) and denote by \(L_p(\mu, X)\) the Banach space of all \(X\)-valued Bochner \(p\)-integrable functions on a certain positive complete \(\sigma\)-finite measure space \((\Omega, \Sigma, \mu)\), endowed with the usual \(p\)-norm. In this paper, the theory of lifting is used to prove that, for any weakly compact subset \(W\) of \(X\), the set \(L_p(\mu, W)\) is \(N\)-simultaneously proximinal in \(L_p(\mu, X)\) for any arbitrary monotonous norm \(N\) in \(\mathbb{R}^n\).