The separable Jung constant in Banach spaces
Volume 258 / 2021
Studia Mathematica 258 (2021), 157-173
MSC: 46B04, 46B20, 46B26, 46M18.
DOI: 10.4064/sm190812-11-5
Published online: 20 November 2020
Abstract
This paper contains a study of the separable version $J_s(\cdot )$ of the classical Jung constant. We first establish, following Davis (1977), that a Banach space $X$ is $1$-separably injective if and only if $J_s(X)=1$. This characterization is then used for the understanding of new $1$-separably injective spaces. The last section establishes the inequality $\frac {1}{2}K(Y)J_s(X)\leq e_1^s(Y,X)$ connecting the separable Jung constant, Kottman’s constant and the separable-one-point extension constant for Lipschitz maps, which is then used to derive an improved version of Kalton’s inequality $K(X,c_0)\leq e(X,c_0)$ and a new characterization of $1$-separable injectivity.
