Uniformly convex space

In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces. The concept of uniform convexity was first introduced by James A. Clarkson in 1936.

Definition

A uniformly convex space is a normed vector space so that, for every $0<\epsilon \leq 2$ there is some $\delta>0$ so that for any two vectors with $\|x\| = 1$ and $\|y\| = 1,$ the condition

$\|x-y\|\geq \epsilon$

implies that:

$\left\|\frac{x+y}{2}\right\| \leq 1-\delta.$

Intuitively, the center of a line segment inside the unit ball must lie deep inside the unit ball unless the segment is short.

Properties

• The Milman–Pettis theorem states that every uniformly convex Banach space is reflexive, while the converse is not true.
• If $\{f_n\}_{n=1}^{\infty}$ is a sequence in a uniformly convex Banach space which converges weakly to $f$ and satisfies $\|f_n\| \to \|f\|,$ then $f_n$ converges strongly to $f$, that is, $\|f_n - f\| \to 0$.
• A Banach space $X$ is uniformly convex if and only if its dual $X^*$ is uniformly smooth.
• Every uniformly convex space is strictly convex.

Examples

• Every Hilbert space is uniformly convex.
• Every closed subspace of a uniformly convex Banach space is uniformly convex.
• Hanner's inequalities imply that Lp spaces $(1 are uniformly convex.
• Conversely, $L^\infty$ is not uniformly convex. For example, in $\mathbb{R}^2$ consider $x=(1,1)$ and $y=(0,1)$. Then $\|x\|_{\infty}=\|y\|_{\infty}=1$ and $\|x+y\|_{\infty}=\|(1,2)\|_{\infty}=2$, but $\|x-y\|_{\infty}=\|(1,0)\|_{\infty}=1$.

• Hanner, O. (1956). "On the uniform convexity of $L^p$ and $l^p$". Ark. Mat. 3: 239–244. doi:10.1007/BF02589410..