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{max1≤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)=max1≤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.