Uniformly convex space explained

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

0<\varepsilon\leq2

there is some

\delta>0

such that for any two vectors with

\|x\|=1

and

\|y\|=1,

the condition

\|x-y\|\geq\varepsilon

implies that:

\left\\|	x+y
	2

\right\|\leq1-\delta.

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

Properties

is uniformly convex if and only if for every

0<\varepsilon\le2

there is some

\delta>0

so that, for any two vectors

and

in the closed unit ball (i.e.

\|x\|\le1

and

\|y\|\le1

) with

\|x-y\|\ge\varepsilon

, one has

\left\\|{	x+y
	2

}\right\| \le 1-\delta (note that, given

\varepsilon

, the corresponding value of

\delta

could be smaller than the one provided by the original weaker definition).The "if" part is trivial. Conversely, assume now that

is uniformly convex and that

x,y

are as in the statement, for some fixed

0<\varepsilon\le2

. Let

\delta_1\le1

be the value of

\delta

corresponding to

	\varepsilon
	3

in the definition of uniform convexity. We will show that

\left\\|	x+y
	2

\right\|\le1-\delta

, with

\delta=min\left\{	\varepsilon	,
	6

	\delta₁
	3

\right\}

\|x\|\le1-2\delta

then

\left\\|	x+y	\right\\|\le
	2

	1	(1-2\delta)+
	2

	1
	2

=1-\delta

and the claim is proved. A similar argument applies for the case

\|y\|\le1-2\delta

, so we can assume that

1-2\delta<\|x\|,\|y\|\le1

. In this case, since

\delta\le	1
	3

, both vectors are nonzero, so we can let

x'=	x
	\\|x\\|

and

y'=	y
	\\|y\\|

. We have

\|x'-x\|=1-\|x\|\le2\delta

and similarly

\|y'-y\|\le2\delta

, so

and

belong to the unit sphere and have distance

\\|x'-y'\\|\ge\\|x-y\\|-4\delta\ge\varepsilon-	4\varepsilon	=
	6

	\varepsilon
	3

. Hence, by our choice of

\delta₁

, we have

\left\\|	x'+y'
	2

\right\|\le1-\delta₁

. It follows that

\left\\|	x+y	\right\\|\le\left\\|
	2

	x'+y'	\right\\|+
	2

	\\|x'-x\\|+\\|y'-y\\|
	2

\le1-\delta_1+2\delta\le1-

	\delta₁
	3

\le1-\delta

and the claim is proved.

The Milman–Pettis theorem states that every uniformly convex Banach space is reflexive, while the converse is not true.
Every uniformly convex Banach space is a Radon–Riesz space, that is, if

\{f_n\}

	infty

	n=1

is a sequence in a uniformly convex Banach space that converges weakly to

and satisfies

\|f_n\|\to\|f\|,

then

f_n

converges strongly to

, that is,

\|f_n-f\|\to0

is uniformly convex if and only if its dual

X^*

is uniformly smooth.

\|x+y\|<\|x\|+\|y\|

whenever

x,y

are linearly independent, while the uniform convexity requires this inequality to be true uniformly.

Examples

Every inner-product space is uniformly convex.^[1]
Every closed subspace of a uniformly convex Banach space is uniformly convex.

(1<p<infty)

are uniformly convex.

Conversely,

L^infty

is not uniformly convex.

References

General references

J. A.. Clarkson. Uniformly convex spaces. Trans. Amer. Math. Soc.. 40. 1936. 396–414. 10.2307/1989630. 1989630. 3. American Mathematical Society. free. .
O.. Hanner. Olof Hanner. On the uniform convexity of
L^p

and
l^p

. Ark. Mat.. 3. 1956. 239–244. 10.1007/BF02589410. free. .
Book: Beauzamy, Bernard. Introduction to Banach Spaces and their Geometry. 1985 . 1982. Second revised. North-Holland. 0-444-86416-4.
10.1007/BF02762802 . Per Enflo. Per Enflo. Banach spaces which can be given an equivalent uniformly convex norm. Israel Journal of Mathematics. 13. 3–4. 1972. 281–288.
Lindenstrauss, Joram and Benyamini, Yoav. Geometric nonlinear functional analysis. Colloquium publications, 48. American Mathematical Society.

Notes and References

Book: Narici . Lawrence . Beckenstein . Edward . Topological Vector Spaces . 2011 . CRC Press . Boca Raton, FL . 978-1-58488-866-6 . 524, Example 16.2.3 . 2nd.