The distance $dist (\mathcal {B},X)$ when $\mathcal {B}$ is a boundary of $B(X^{**})$

Abstract:

Let $X$ be a real Banach space and let $\mathcal {B}$ be a boundary of the unit ball $B(X^{**})$ of the bidual $X^{**}$ (which means that for each $x^*\in X^*$ there is $b\in \mathcal {B}$ such that $\langle b,x^*\rangle =\|x^*\|$). We show that $dist(\mathcal {B},X)=dist(B(X^{**}),X)$ where $dist(A,X)$ denotes the sup of all $dist(a, X)$ with $a\in A$. Since $\overline {\mathrm {co}}^{w^*}(\mathcal {B})=B(X^{**})$ this is in contrast with the fact that in general strict inequality can occur between $dist(K,X)$ and $dist(\overline {\mathrm {co}}^{w^*}(K),X)$ even for a $w^*$-compact $K\subset X^{**}$.