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Journal of Convex Analysis 13 (2006), No. 3, 711--719
Copyright Heldermann Verlag 2006

Weak-Star Convergence of Convex Sets

Simon P. Fitzpatrick

Adrian S. Lewis
ORIE, Cornell University, Ithaca, NY 14853, U. S. A.


We show that if a Banach space $X$ is weakly compactly generated and $C$, $C_n$ are weak-star-closed bounded convex nonempty subsets of the dual space $X^*$, then the support functionals $\delta^*_{C_n}$ converge to $\delta^*_C$ pointwise on $X$ if and only if the sequence $(C_n)$ is uniformly bounded with weak-star limit $C$.

Keywords: Scalar convergence, weak-star convergence, set convergence, weakly compactly generated.

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