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! Cheers.