Distances to convex sets

Antonio S. Granero ¹   ; Marcos Sánchez ¹

¹ Departamento de Análisis Matemático Facultad de Matemáticas Universidad Complutense de Madrid 28040 Madrid, Spain

Studia Mathematica, Tome 182 (2007) no. 2, pp. 165-181

Résumé

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 $C$.

Détail

DOI : 10.4064/sm182-2-5

Keywords: banach space convex subset nbsp * investigate whether distance hat overline * sup inf k c overline * overline * m controlled distance hat hat overline * leq hat leq infty weak * compact subset * prove example has control contains copy basis ell has control subset subset * subspace weak * angelic closed dual unit ball * iii convex subset considered canonically embedded its bidual ** has control inside ** general control cap weak * dense