Strong martingale type and uniform smoothness
  We introduce stronger versions of the usual notions of martingale type p <= 2 and cotype q >= 2 of a Banach space X and show that these concepts are equivalent to uniform p-smoothness and q-convexity, respectively. All these are metric concepts, so they depend on the particular norm in X.   These concepts allow us to get some more insight into the fine line between X being isomorphic to a uniformly p-smooth space or being uniformly p-smooth itself.   Instead of looking at Banach spaces, we consider linear operators between Banach spaces right away. The situation of a Banach space X can be rediscovered from this by considering the identity map of X. 
J\"org Wenzel