Resumen de The spaces of compact convex sets and bounded closed convex sets in a Banach space

Katsuro Sakai

• Let X be an infinite-dimensional Banach space with density t and let CC(X) and BC(X) be the spaces of all non-empty compact convex sets in X and of all non-empty bounded closed convex sets admitting the Hausdorff metric, respectively. In this note, it is proved that (i) the space CC(X) is homeomorphic to the Hilbert space with density t; (ii) the space BC(X) is homeomorphic to the Hilbert space with density t if it has the density t; (iii) the space BC(X) is homeomorphic to the Hilbert space with density 2t if X is separable (i.e., t is countable) or reflective