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On sharp uniform convexity, smoothness, and strong type, cotype inequalities. (English) Zbl 1030.46012
In the first part the authors show that certain modifications of Clarkson’s inequality are equivalent to the definition of $$p$$-uniform smoothness and $$q$$-uniform convexity of a Banach space $$X$$. They also show duality for these modifications.
Then the concepts of $$p$$-uniform smoothness and $$q$$-uniform convexity are shown to be equivalent to certain variants of Rademacher type $$p$$ and cotype $$q$$ inequalities, dubbed strong type $$p$$ and strong cotype $$q$$. The authors study duality of strong type and cotype and how these properties pass from $$X$$ to $$L_p(X)$$.
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)].

##### MSC:
 46B20 Geometry and structure of normed linear spaces 46B07 Local theory of Banach spaces 46E40 Spaces of vector- and operator-valued functions 46B04 Isometric theory of Banach spaces