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Let $C$ be the class of Banach spaces $X$ such that there exists $0<\theta<1$, a Hilbert space $H$ and a Banach space $Y$ such that $$ X=(H,Y)_\theta $$ (complex interpolation of Calderon).

Does there exist a geometric charaterization of this class of Banach spaces?

I know that a Banach space $X$ in $C$ is uniformly convex.

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