I would like to find an example of a reflexive Banach space $X$, which is locally uniformly convex, however, it is not uniformly convex.
The motivation is that I am studying degree theory for operators $T:X\to X^\star$, which satisfies the $S_+$ condition and in some point, it is assumed that $X$ can be renormed in such a way that the new norm is equivalent to the old one and both $X,X^\star$ are locally uniformly convex (Lindenstrauss-Asplund-Trojanski theorem).
It seems that the example is totally non-trivial. Any reference is appreciated.