On Some Properties of Hyperconvex Spaces

Abstract

We are going to answer some open questions in the theory of hyperconvex metric spaces. We prove that in complete ℝ-trees hyperconvex hulls are uniquely determined. Next we show that hyperconvexity of subsets of normed spaces implies their convexity if and only if the space under consideration is strictly convex. Moreover, we prove a Krein-Milman type theorem for ℝ-trees. Finally, we discuss a general construction of certain complete metric spaces. We analyse its particular cases to investigate hyperconvexity via measures of noncompactness.