A NONLINEAR MAP FOR MIDPOINT LOCALLY UNIFORMLY ROTUND RENORMING

We provide a criterion for midpoint locally uniformly rotund renormability of normed spaces involving the class of -slicely continuous maps, recently introduced by Molto, Orihuela, Troyanski and Valdiva in 2003. As a consequence of this result, we obtain a theorem of G. Alexandrov concerning the three space problem for midpoint locally uniformly rotund renormings of Banach spaces. A normed space (X,k · k) (or its norm) is said to be midpoint locally uniformly rotund if for every x 2 X and every sequence (xn)n in X such that kxn + xk!kxk and kxn xk!kxk we have kxnk!0. Recall also that X is locally uniformly rotund if for every x 2 X and every sequence (xn)n X such that lim n kxnk = kxk and lim n kxn +xk = 2kxk we have lim n kxn xk = 0, and that X is strictly convex or rotund (R for short) if x = y whenever x and y are elements of X such that kxk = kyk = (x + y)/2 . It is clear that locally uniformly rotund ) midpoint locally uniformly rotund and that midpoint locally uniformly rotund ) R. In the paper [5], devoted to the renorming of spaces of continuous functions on trees, R. Haydon provides the first example (the only known to date) of midpoint locally uniformly rotund space with no equivalent locally uniformly rotund renorming. There, he also shows that for every tree , the existence of an equivalent strictly convex norm on C() implies midpoint locally uniformly rotund renormability on this space. This coincidence is not true in general: an example of strictly convexifiable space without midpoint locally uniformly rotund renorming is ‘1 (see [2, 6]). Our aim in this paper is to provide a criterion for midpoint locally uniformly rotund renorming of spaces that have images in midpoint locally uniformly rotund spaces through special non linear maps. These are the -slicely continuous maps recently introduced in [11], where a non linear transfer technique for locally uniformly rotund renormability has been developed.