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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[edit]

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[edit]

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[edit]

Every inner-product space is uniformly convex.^[1]
Every closed subspace of a uniformly convex Banach space is uniformly convex.
Clarkson's inequalities imply that L^p spaces $(1<p<\infty )$ are uniformly convex.
Conversely, $L^{\infty }$ is not uniformly convex.

References[edit]

Citations[edit]

^ 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[edit]

Clarkson, J. A. (1936). "Uniformly convex spaces". Trans. Amer. Math. Soc. 40 (3). American Mathematical Society: 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.

[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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