# 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 ${\displaystyle 0<\epsilon \leq 2}$ there is some ${\displaystyle \delta >0}$ so that for any two vectors with ${\displaystyle \|x\|=1}$ and ${\displaystyle \|y\|=1,}$ the condition

${\displaystyle \|x-y\|\geq \epsilon }$

implies that:

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

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

## References

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• Lindenstrauss, Joram and Benyamini, Yoav. Geometric nonlinear functional analysis Colloquium publications, 48. American Mathematical Society.