TITLE: Are Sobolev Spaces Uniformly Convex?
QUESTION [6 upvotes]: Let $W^{m,p}(\Omega)$ be a Sobolev space, where $\Omega\subset\mathbb{R}^{n}$ is a open set.
So the question is: For what $p$ and $m$ the space $W^{m,p}(\Omega)$ is Uniformly Convex.
Some reference or the answer would be appreciated.

REPLY [9 votes]: Note, that the map ($N := |\{\alpha \in \mathbb N_{\ge 0}^n \mid  |\alpha|\le m\}|$)
\begin{align*}
  T\colon W^{m,p}(\Omega) &\to L^p(\Omega, \mathbb R^{N})\\
                  x &\mapsto (D^\alpha x)_{|\alpha| \le m}
\end{align*}
is a closed and isometric embedding. As $L^p(\Omega, \mathbb R^N)$ is uniformly convex for $1 < p < \infty$, so are its closed subspaces, and hence as $W^{m,p}(\Omega)$ is isometric to its image under $T$, so is $W^{m,p}(\Omega)$ for these $p$. 
On the other side, uniformly convex spaces are reflexive and $W^{m,1}(\Omega)$ and $W^{m,\infty}(\Omega)$ aren't.