A Banach space $X$ is isomorphic to a uniformly convex space if and only if the diamond graphs $D_n$ Lipschitz do not embed into $X$ with distortions independent of $n$. The same holds for the Laakso graphs.
One of the consequences of that and previously known results is that dimension reduction a-la Johnson--Lindenstrauss fails in any non uniformly convex Banach space with non trivial type (it is known that there are such spaces).
Joint work with Bill Johnson.
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