# Uniformly convex space

Uniformly convex space

In mathematics, uniformly convex spaces are common examples of reflexive Banach spaces. These include all Hilbert spaces and the L"p" spaces for

Definition

A uniformly convex space is a Banach space so that, for every $epsilon>0$ there is some $delta>0$ so that for any two vectors with $|x|le1$ and $|y|le 1,$

:$|x+y|>2-delta$

implies

:

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 space is reflexive.

ee also

* Modulus and characteristic of convexity
* Hanner's inequalities say that L"p" spaces

References

*.
*.

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