Convex -closures versus convex norm-closures in dual Banach spaces

A subset Y of a dual Banach space X ∗ is said to have the property ( P ) if co ¯ w ∗ ( H ) = co ¯ ( H ) for every weak * -compact subset H of Y. The purpose of this paper is to give a characterization of the property ( P ) for subsets of a dual Banach space X ∗ , and to study the behavior of the property ( P ) with respect to additions, unions, products, whether the closed linear hull [ Y ] ¯ has the property ( P ) when Y does, etc. We show that the property ( P ) is stable under all these operations in the class of weak * K -analytic subsets of X ∗ .