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
0 < ε ≤ 2
δ > 0
such that for any two vectors with
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.
The "if" part is trivial.
Conversely, assume now that
is uniformly convex and that
are as in the statement, for some fixed
0 < ε ≤ 2
δ
ε 3
in the definition of uniform convexity.
We will show that
δ = min
ε 6
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 similarly
belong to the unit sphere and have distance
‖ ≥ ‖ x − y ‖ − 4 δ ≥ ε −
4 ε
ε 3
Hence, by our choice of
and the claim is proved.