uniformly convex space

A normed space is uniformly convex iff $\forall ϵ>0$ there exists $\delta >0$ that satisfiesfor $\parallel x\parallel \le 1$ $\parallel y\parallel \le 1$ and $\parallel x-y\parallel >ϵ$ $\Rightarrow$ $\parallel \frac{x+y}{2}\parallel \le 1$-$\delta$.

For example it is easily seen that the normed space $({ℝ}^2,\parallel .{\parallel }_2)$ is uniformly convex space.Also $L^p$ and $l^p$ spaces for are uniformly convex, see J.A. Clarkson, ”Uniformly convex spaces”, Trans. Amer. Math. Society, 40 (1936), 396-414.