Local U-Convexity

K.-S. Lau ["Best approximation by closed sets in Banach spaces", J. Approx. Theory 23 (1978) 29--36] considered the notion of "U-convex spaces" (originally called U-spaces) and showed that both uniform convexity and uniform smoothness imply U-convexity. Also U-convex spaces are uniformly non-square and hence super-reflexive. In this paper we introduce local U-convexity. It is shown that there are two possible localization of U-convexity. 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.