In the first part the authors show that certain modifications of Clarkson’s inequality are equivalent to the definition of -uniform smoothness and -uniform convexity of a Banach space . They also show duality for these modifications.
Then the concepts of -uniform smoothness and -uniform convexity are shown to be equivalent to certain variants of Rademacher type and cotype inequalities, dubbed strong type and strong cotype . The authors study duality of strong type and cotype and how these properties pass from to .
All this continues and extends previous work by the authors M. Kato, L.-E. Persson, and Y. Takahashi [Collect. Math. 51, 327-346 (2000; Zbl 0983.46014)] and M. Kato and Y. Takahashi [Math. Nachr. 186, 187-195 (1997; Zbl 0901.46013)].