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Let $X$ be a Banach space. Suppose that $X$ has UMD (hence super-refexive). It seems to me that the following equivalence is classical.
Then $X$ is of cotype $q$-uniformly convex if and only if $X$ is of cotype $q$-uniformly convex.
Let $X$ be a Banach space. Suppose that $X$ has UMD. It seems to me that the following equivalence is classical.
Then $X$ is $q$-uniformly convex if and only if $X$ is of cotype $q$.
Then $X$ is of cotype $q$ if and only if $X$ is $q$-uniformly convex.
