Contracting the maximal points of an ordered convex set

The maximal points of a nonempty closed bounded convex set in a reflexive Banach spare, relative to an ordering defined by a locally uniformly convex cone, are studied. The set of maximal points is proved to be contractible, and sufficient conditions are found for it to be contractible by a homotopy with the semigroup property, or by the flow of an ordinary differential equation.