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.
A uniformly convex space is a normed vector space such that, for every there is some such that for any two vectors with and 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.
Proof
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The "if" part is trivial. Conversely, assume now that is uniformly convex and that are as in the statement, for some fixed . Let be the value of 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 and belong to the unit sphere and have distance . Hence, by our choice of , we have . It follows that and the claim is proved. |