  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.
