Abstract
We are going to answer some open questions in the theory of hyperconvex metric spaces. We prove that in complete -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 -trees. Finally, we discuss a general construction of certain complete metric spaces. We analyse its particular cases to investigate hyperconvexity via measures of noncompactness.
Highlights
It is hard to believe that hyperconvex metric spaces have been investigated for more that fifty years, some basic questions in their theory still remain open let us recall that hyperconvex metric spaces were introduced in 1 see 2, but from formal point of view it has to be emphasized that the notion of hyperconvexity was investigated earlier by Aronszajn in his Ph.D. thesis 3 which was never published
Let us begin with the notion of hyperconvex hull which was introduced by Isbell in 4 see Definition 2.7
Let us recall that such hyperconvex spaces were characterized by Kirk see 5 as complete R-trees see Theorem 2.15
Summary
It is hard to believe that hyperconvex metric spaces have been investigated for more that fifty years, some basic questions in their theory still remain open let us recall that hyperconvex metric spaces were introduced in 1 see 2 , but from formal point of view it has to be emphasized that the notion of hyperconvexity was investigated earlier by Aronszajn in his Ph.D. thesis 3 which was never published. We prove that a bounded complete R-tree is a convex hull of its extremal points note that a similar result, but with the assumption of compactness, is proved in 9. In particular, such a property holds for bounded hyperconvex metric spaces with unique metric segments. Assume that α A 2β A for every bounded subset of a given metric space X Does this equality imply that X is hyperconvex?
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