TITLE: A reflexive space which does not have an equivalent uniformly convex norm
QUESTION [14 upvotes]: I found this beautiful theorem due to Milman and Pettis:

Every uniformly convex Banach space is reflexive.

I think it's a remarkable statement, since uniformly convexity is a geometric property of the norm and therefore need not to be true for an equivalent norm.
While reflexivity is a topological statement and therefore a reflexive space remains reflexive for an equivalent norm. 
My first question is: 

Could someone give an example of a space with two equivalent norms such that the space is uniformly convex for just one of the two norms. 

And my second question is: 

Could someone give me an example of a reflexive space that admits no uniformly convex equivalent norm.

Thank you in advance for answering my questions!

REPLY [4 votes]: Note that every separable space can be given an equivalent strictly convex norm. Indeed, if $X$ is separable then you can easily construct an injective operator $T\colon X\to \ell_2$. Then define a new norm on $X$ by $\|x\|^\prime = \|x\|+\|Tx\|_{\ell_2}$, which is strictly convex. (I guess this trick is attributed to Victor Klee.)
Now, to answer your second question we need a reflexive space which is separable but not super-reflexive because then we can endow it with a strictly convex norm, yet it cannot be uniformly convex. (Actually, every reflexive space can be given an equivalent strictly convex norm but this is more involved.) For this take
$$X = \big( \bigoplus_{n=1}^\infty \ell_1^n \big)_{\ell_2},$$
where $\ell_1^n$ stands for $\mathbb{R}^n$ endowed with the $\ell_1$-norm. By Enflo's thoerem, it cannot be given an equivalent uniformly convex norm because it is not super-reflexive. Of course, Enflo's theorem characterising super-reflexive spaces as precisely those which can be renormed in a uniformly convex manner is very deep.