$L^p $ is not uniformly convex for $p=1, \infty$

Question

As the title says: how can we prove that $L^p(\mathbb R^n)$ is not uniformly convex for $p=1$ and $p=\infty$.

Does anyone knows a counter-example for the cases $ p=1$ and $ p = \infty$ for the space $L^p( \mathbb R^n)$ ?

I am studying N.L. Carothers book "A short course to Banach space theory".

Any help?

Answer 1

For $p=\infty$ and $n=2$, consider $\mathbf{x}=(1,1)$ and $\mathbf{y}=(0,1)$.

You might also make use of the fact that uniformly convex Banach spaces are reflexive.

Edit: As I was typing up a more detailed answer, I found the first comment to this answer, which puts things more succinctly than I was going to do (and saves me a bit of typing).