Uniformly convex spaces

[to be updated …]

Every uniformly convex space is reflexive, see Proposition 2.4.11. page 55 [1]. So,
$$\text{Hilbert} \Longrightarrow \text{uniformly convex} \Longrightarrow \text{reflexive}$$

Uniformly characterization

It is well known that the Hilbert space is uniformly convex due to the parallelogram
equality:
$$||u+v||^2+||u-v||^2=2(||u||^2+||v||^2)$$

Similarly, the $L^p$ spaces for $p\in (1, +\infty)$ is also uniformly convex due the inequality called Clarkson’s inequalities, see Proposition 2.4.9. page 53 [1],

• For $p\in [2, +\infty)$, $||\frac{u+v}{2}||^p+||\frac{u-v}{2}||^p \leq \frac{1}{2}(||u||^p+||v||^p)$
• For $p\in (1, 2]$, $||\frac{u+v}{2}||^{p'}+||\frac{u-v}{2}||^{p'} \leq \frac{1}{2}(||u||^{p'}+||v||^{p'})^{p-1}$ where $\frac{1}{p}+\frac{1}{p'}=1$.

References

[1]: Attouch, Hedy, Giuseppe Buttazzo, and Gérard Michaille. Variational analysis in Sobolev and BV spaces: applications to PDEs and optimization. Society for Industrial and Applied Mathematics, 2014.