On Some Properties of Hyperconvex Spaces

We are going to answer some open questions in the theory of hyperconvex metric spaces. We prove that in complete [inline-graphic not available: see fulltext]-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 [inline-graphic not available: see fulltext]-trees. Finally, we discuss a general construction of certain complete metric spaces. We analyse its particular cases to investigate hyperconvexity via measures of noncompactness.