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 such that, for every $0<\varepsilon \leq 2$ 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\|{\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.

Properties

The unit sphere can be replaced with the closed unit ball in the definition. Namely, a normed vector space $X$ is uniformly convex if and only if for every $0<\varepsilon \leq 2$ there is some $\delta >0$ so that, for any two vectors $x$ and $y$ in the closed unit ball (i.e. $\|x\|\leq 1$ and $\|y\|\leq 1$ ) with $\|x-y\|\geq \varepsilon$ , one has $\left\|{\frac {x+y}{2}}\right\|\leq 1-\delta$ (note that, given $\varepsilon$ , the corresponding value of $\delta$ could be smaller than the one provided by the original weaker definition).

Proof

The "if" part is trivial. Conversely, assume now that $X$ is uniformly convex and that $x,y$ are as in the statement, for some fixed $0<\varepsilon \leq 2$ . Let $\delta _{1}\leq 1$ be the value of $\delta$ corresponding to ${\frac {\varepsilon }{3}}$ in the definition of uniform convexity. We will show that $\left\|{\frac {x+y}{2}}\right\|\leq 1-\delta$ , with $\delta =\min \left\{{\frac {\varepsilon }{6}},{\frac {\delta _{1}}{3}}\right\}$ .

If $\|x\|\leq 1-2\delta$ then $\left\|{\frac {x+y}{2}}\right\|\leq {\frac {1}{2}}(1-2\delta )+{\frac {1}{2}}=1-\delta$ and the claim is proved. A similar argument applies for the case $\|y\|\leq 1-2\delta$ , so we can assume that $1-2\delta <\|x\|,\|y\|\leq 1$ . In this case, since $\delta \leq {\frac {1}{3}}$ , both vectors are nonzero, so we can let $x'={\frac {x}{\|x\|}}$ and $y'={\frac {y}{\|y\|}}$ . We have $\|x'-x\|=1-\|x\|\leq 2\delta$ and similarly $\|y'-y\|\leq 2\delta$ , so $x'$ and $y'$ belong to the unit sphere and have distance $\|x'-y'\|\geq \|x-y\|-4\delta \geq \varepsilon -{\frac {4\varepsilon }{6}}={\frac {\varepsilon }{3}}$ . Hence, by our choice of $\delta _{1}$ , we have $\left\|{\frac {x'+y'}{2}}\right\|\leq 1-\delta _{1}$ . It follows that $\left\|{\frac {x+y}{2}}\right\|\leq \left\|{\frac {x'+y'}{2}}\right\|+{\frac {\|x'-x\|+\|y'-y\|}{2}}\leq 1-\delta _{1}+2\delta \leq 1-{\frac {\delta _{1}}{3}}\leq 1-\delta$ 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 $\{f_{n}\}_{n=1}^{\infty }$ is a sequence in a uniformly convex Banach space that converges weakly to $f$ and satisfies $\|f_{n}\|\to \|f\|,$ then $f_{n}$ converges strongly to $f$ , that is, $\|f_{n}-f\|\to 0$ .
A Banach space $X$ is uniformly convex if and only if its dual $X^{*}$ is uniformly smooth.
Every uniformly convex space is strictly convex. Intuitively, the strict convexity means a stronger triangle inequality $\|x+y\|<\|x\|+\|y\|$ 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.
Hanner's inequalities imply that L^p spaces $(1<p<\infty )$ are uniformly convex.
Conversely, $L^{\infty }$ is not uniformly convex.

References

Citations

↑ Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces (2nd ed.). Boca Raton, FL: CRC Press. p. 524, Example 16.2.3. ISBN 978-1-58488-866-6.

General references

Clarkson, J. A. (1936). "Uniformly convex spaces". Trans. Amer. Math. Soc. American Mathematical Society. 40 (3): 396–414. doi:10.2307/1989630. JSTOR 1989630..
Hanner, O. (1956). "On the uniform convexity of $L^{p}$ and $l^{p}$ ". Ark. Mat. 3: 239–244. doi:10.1007/BF02589410..
Beauzamy, Bernard (1985) [1982]. Introduction to Banach Spaces and their Geometry (Second revised ed.). North-Holland. ISBN 0-444-86416-4.
Per Enflo (1972). "Banach spaces which can be given an equivalent uniformly convex norm". Israel Journal of Mathematics. 13 (3–4): 281–288. doi:10.1007/BF02762802.
Lindenstrauss, Joram and Benyamini, Yoav. Geometric nonlinear functional analysis. Colloquium publications, 48. American Mathematical Society.

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[1] Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces (2nd ed.). Boca Raton, FL: CRC Press. p. 524, Example 16.2.3. ISBN 978-1-58488-866-6.

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