Finite slicing in superreflexive Banach spaces

For every superreflexive Banach space X there exists a supermultiplicative function which is the supremum, in a very natural ordering, of the set of all the moduli of convexity of equivalent norms. If this supremum is actually a maximum achieved under some equivalent renorming of X, then its modulus of convexity is the best possible in asymptotic sense. Otherwise, we can give an almost optimal uniformly convex renorming of X beyond the classical power type bound obtained by Pisier [21].