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