Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 213812, 19 pages
doi:10.1155/2010/213812
On some properties of hyperconvex spaces
Marcin Borkowski1
, Dariusz Bugajewski2
and Dev Phulara3
1Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland
2Department of Mathematics, Morgan State University, 1700 E. Cold Spring Lane, Baltimore, MD 21251, USA
3Department of Mathematics, Howard University, 2400 Sixth Street, NW, Washington, DC 20059, USA
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.