On Simultaneous Farthest Points in L∞(I, X)

Abstract

Let X be a Banach space and let G be a closed bounded subset of X. For (x1, x2, …, xm) ∈ Xm, we set ρ(x1, x2, …, xm, G) = sup {max 1≤i≤m ∥xi − y∥ : y ∈ G}. The set G is called simultaneously remotal if, for any (x1, x2, …, xm) ∈ Xm, there exists g ∈ G such that ρ(x1, x2, …, xm, G) = max 1≤i≤m ∥xi − g∥. In this paper, we show that if G is separable simultaneously remotal in X, then the set of ∞‐Bochner integrable functions, L∞(I, G), is simultaneously remotal in L∞(I, X). Some other results are presented.