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 k-super-strongly smooth spaces. It is proved that these two notions are dual. We also prove that the class of k-super-strongly convexitiable 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 greater than or equal to 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. (C) 2004 Elsevier Inc. All rights reserved.