The separable Jung constant in Banach spaces

TL;DR

This paper investigates the separable Jung constant in Banach spaces, characterizes 1-separably injective spaces, and establishes inequalities linking various geometric constants, providing new insights and simpler proofs in Banach space theory.

Contribution

It characterizes 1-separably injective Banach spaces via the separable Jung constant and connects this constant to Kottman's and extension constants, offering new characterizations and simpler proofs.

Findings

Characterization of 1-separably injective spaces using the separable Jung constant.

Establishment of an inequality linking the separable Jung constant, Kottman's constant, and extension constants.

Simplified proof of the equality between Kottman's constant and the extension constant for c0.

Abstract

This paper contains a study of the separable form $J_s(\cdot)$ of the classical Jung constant. We first establish, following Davis \cite{davis}, 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 $\frac{1}{2}K(Y)J_s(X)\leq e(Y,X)$ connecting the separable Jung constant, Kottman's constant and the extension constant for Lipschitz maps, which is then used to obtain a simple proof of the equality $K(X,c_0)=e(X,c_0)$ of Kalton and a new characterization of $1$-separable injectivity.