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 such that, for every there is some \delta>0 such that for any two vectors with |x| = 1 and |y| = 1, the condition implies that: 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 "if" part is trivial. Conversely, assume now that X is uniformly convex and that x,y are as in the statement, for some fixed. Let be the value of \delta corresponding to in the definition of uniform convexity. We will show that, with. If then and the claim is proved. A similar argument applies for the case, so we can assume that. In this case, since, both vectors are nonzero, so we can let and. We have and similarly, so x' and y' belong to the unit sphere and have distance. Hence, by our choice of \delta_1, we have. It follows that and the claim is proved.

Examples

Citations

General references