
Journal of Convex Analysis 18 (2011), No. 3, 811821 Copyright Heldermann Verlag 2011 Local UConvexity Sudipta Dutta Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, India sudipta@iitk.ac.in BorLuh Lin Dept. of Mathematics, University of Iowa, Iowa City, U.S.A. bllin@math.uiowa.edu K.S. Lau ["Best approximation by closed sets in Banach spaces", J. Approx. Theory 23 (1978) 2936] considered the notion of "Uconvex spaces" (originally called Uspaces) and showed that both uniform convexity and uniform smoothness imply Uconvexity. Also Uconvex spaces are uniformly nonsquare and hence superreflexive. In this paper we introduce local Uconvexity. It is shown that there are two possible localization of Uconvexity. We derive our results quantitatively, that is, by the properties of modulus functions. Relationship to modulus of (local) uniform convexity is established and its consequences are discussed. Keywords: Locally uniformly convex, superreflexive spaces, Uconvexity. MSC: 46B20 [ Fulltextpdf (141 KB)] for subscribers only. 