Distances to convex sets
Volume 182 / 2007
Abstract
If $X$ is a Banach space and $C$ a convex subset of $X^*$, we investigate whether the distance $\hat d({{\overline {\rm {co}}}}^{w^*}(K),C):=\sup \{\inf\{\|k-c\|:c\in C\}:k\in \overline {\rm {co}} ^{w^*}(K)\}$ from $\overline {\rm {co}} ^{w^*}(K)$ to $C$ is $M$-controlled by the distance $\hat d(K,C)$ (that is, if $\hat d({{\overline {\rm {co}}}}^{w^*}(K),C)\leq M \hat d(K,C)$ for some $1\leq M<\infty $), when $K$ is any weak$^*$-compact subset of $X^*$. We prove, for example, that: (i) $C$ has 3-control if $C$ contains no copy of the basis of $\ell _1( c )$; (ii) $C$ has 1-control when $C\subset Y\subset X^*$ and $Y$ is a subspace with weak$^*$-angelic closed dual unit ball $B(Y^*)$; (iii) if $C$ is a convex subset of $X$ and $X$ is considered canonically embedded into its bidual $X^{**}$, then $C$ has 5-control inside $X^{**}$, in general, and 2-control when $K\cap C$ is weak$^*$-dense in $C$.
