ON UNIFORMLY CONVEX AND UNIFORMLY KADEC-KLEE RENORMINGS

Abstract

We give a new construction of uniformly convex norms with a power type modulus on super-reflexive spaces based on the notion of dentability index. Furthermore, we prove that if the Szlenk index of a Banach space is less than or equal to ω (first infinite ordinal) then there is an equivalent weak ∗ lower semi-continuous positively homogeneous functional on X ∗ satisfying the uniform Kadec-Klee Property for the weak∗-topology (UKK∗). Then we solve the UKK or UKK∗ renorming problems for Lp(X) spaces and C(K) spaces for K scattered compact space.