k-super-strongly convex and k-super-strongly smooth Banach spaces ✩

In this paper, we introduce two types of new Banach spaces: k-super-strongly convex spaces and ksuper- strongly smooth spaces. It is proved that these two notions are dual.We also prove that the class of k-super-strongly convexifiable spaces is strictly between locally k-uniformly rotund spaces and k-strongly convex spaces, and obtain some necessary and sufficient conditions of k-super-strongly convex space (respectively k-super-strongly smooth space). Also, for each k 2, it is shown that there exists a k-super-strongly convex (respectively k-super-strongly smooth) space which is not (k − 1)-super-strongly convex (respectively (k −1)-super-strongly smooth) space.  2004 Elsevier Inc. All rights reserved.