# uniformly convex space

A normed space is uniformly convex iff $\forall\epsilon>0$ there exists $\delta>0$ that satisfies for $\|x\|\leq 1$ $\|y\|\leq 1$ and $\|x-y\|>\epsilon$ $\Rightarrow$ $\|\frac{x+y}{2}\|\leq 1$-$\delta$.

For example it is easily seen that the normed space $(\mathbb{R}^{2},\|.\|_{2})$ is uniformly convex space. Also $L^{p}$ and $l^{p}$ spaces for $1 are uniformly convex, see J.A. Clarkson, ”Uniformly convex spaces”, Trans. Amer. Math. Society, 40 (1936), 396-414.

Title uniformly convex space UniformlyConvexSpace 2013-03-22 15:13:11 2013-03-22 15:13:11 georgiosl (7242) georgiosl (7242) 32 georgiosl (7242) Definition msc 46H05 uniformly convex