On k-strong convexity in banach spaces

We introduce and study the notion of k-strong convexity in Banach spaces. It is a generalization of the notion of strong convexity first studied by Fan and Glicksberg. A Banach space is said to be k-strongly convex if it is reflexive, k-strictly convex and has the Kadec-Klee property. We use the idea of k-dimensional diameter to give several characterizations of k-strong convexity. Further, we study k-strict convexity and k-strong convexity in some products of Banach spaces. Finally, we give characterizations of k-uniform convexity that distinguish it from k-strong convexity.