Uniformly convex space

A uniformly convex space is a normed vector space such that, for every there is some $\backslash delta>0$ such that for any two vectors with $\backslash |x\backslash |\; =\; 1$ and $\backslash |y\backslash |\; =\; 1,$ the condition

:$\backslash |x-y\backslash |\backslash geq\backslash varepsilon$

implies that:

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

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

• 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 there is some $\backslash delta>0$ so that, for any two vectors $x$ and $y$ in the closed unit ball (i.e. $\backslash |x\backslash |\; \backslash le\; 1$ and $\backslash |y\backslash |\; \backslash le\; 1$) with $\backslash |x-y\backslash |\; \backslash ge\; \backslash varepsilon$, one has $\backslash left\backslash |\{\backslash frac\{x+y\}\{2\}\}\backslash right\backslash |\; \backslash le\; 1-\backslash delta$ (note that, given $\backslash varepsilon$, the corresponding value of $\backslash delta$ could be smaller than the one provided by the original weaker definition).

{{Collapsed top|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 . Let $\backslash delta\_1\backslash le\; 1$ be the value of $\backslash delta$ corresponding to $\backslash frac\{\backslash varepsilon\}\{3\}$ in the definition of uniform convexity. We will show that $\backslash left\backslash |\backslash frac\{x+y\}\{2\}\backslash right\backslash |\backslash le\; 1-\backslash delta$, with $\backslash delta=\backslash min\backslash left\backslash \{\backslash frac\{\backslash varepsilon\}\{6\},\backslash frac\{\backslash delta\_1\}\{3\}\backslash right\backslash \}$.

If $\backslash |x\backslash |\backslash le\; 1-2\backslash delta$ then $\backslash left\backslash |\backslash frac\{x+y\}\{2\}\backslash right\backslash |\backslash le\backslash frac\{1\}\{2\}(1-2\backslash delta)+\backslash frac\{1\}\{2\}=1-\backslash delta$ and the claim is proved. A similar argument applies for the case $\backslash |y\backslash |\backslash le\; 1-2\backslash delta$, so we can assume that . In this case, since $\backslash delta\backslash le\backslash frac\{1\}\{3\}$, both vectors are nonzero, so we can let $x\text{'}=\backslash frac\{x\}\{\backslash |x\backslash |\}$ and $y\text{'}=\backslash frac\{y\}\{\backslash |y\backslash |\}$. We have $\backslash |x\text{'}-x\backslash |=1-\backslash |x\backslash |\backslash le\; 2\backslash delta$ and similarly $\backslash |y\text{'}-y\backslash |\backslash le\; 2\backslash delta$, so $x\text{'}$ and $y\text{'}$ belong to the unit sphere and have distance $\backslash |x\text{'}-y\text{'}\backslash |\backslash ge\backslash |x-y\backslash |-4\backslash delta\backslash ge\backslash varepsilon-\backslash frac\{4\backslash varepsilon\}\{6\}=\backslash frac\{\backslash varepsilon\}\{3\}$. Hence, by our choice of $\backslash delta\_1$, we have $\backslash left\backslash |\backslash frac\{x\text{'}+y\text{'}\}\{2\}\backslash right\backslash |\backslash le\; 1-\backslash delta\_1$. It follows that $\backslash left\backslash |\backslash frac\{x+y\}\{2\}\backslash right\backslash |\backslash le\backslash left\backslash |\backslash frac\{x\text{'}+y\text{'}\}\{2\}\backslash right\backslash |+\backslash frac\{\backslash |x\text{'}-x\backslash |+\backslash |y\text{'}-y\backslash |\}\{2\}\backslash le\; 1-\backslash delta\_1+2\backslash delta\backslash le\; 1-\backslash frac\{\backslash delta\_1\}\{3\}\backslash le\; 1-\backslash delta$ and the claim is proved.

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• 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 $\backslash \{f\_n\backslash \}\_\{n=1\}^\{\backslash infty\}$ is a sequence in a uniformly convex Banach space that converges weakly to $f$ and satisfies $\backslash |f\_n\backslash |\; \backslash to\; \backslash |f\backslash |,$ then $f\_n$ converges strongly to $f$, that is, $\backslash |f\_n\; -\; f\backslash |\; \backslash to\; 0$.
• 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 whenever $x,y$ are linearly independent, while the uniform convexity requires this inequality to be true uniformly.

• Every inner-product space is uniformly convex.{{cite book |last1=Narici |first1=Lawrence |last2=Beckenstein |first2=Edward |title=Topological Vector Spaces |date=2011 |publisher=CRC Press |location=Boca Raton, FL |isbn=978-1-58488-866-6 |page=524, Example 16.2.3 |edition=2nd}}
• Every closed subspace of a uniformly convex Banach space is uniformly convex.
• Clarkson's inequalities imply that L^p spaces
• Conversely, $L^\backslash infty$ is not uniformly convex.

• Modulus and characteristic of convexity
• Uniformly convex function
• Uniformly smooth space

{{reflist}}

• {{Cite journal|first=J. A.|last=Clarkson|title=Uniformly convex spaces|journal=Trans. Amer. Math. Soc.|volume=40|year=1936|pages=396–414|doi=10.2307/1989630|jstor=1989630|issue=3|publisher=American Mathematical Society|doi-access=free}}.
• {{Cite journal|first=O.|last=Hanner|author-link=Olof Hanner|title=On the uniform convexity of $L^p$ and $l^p$|journal=Ark. Mat.|volume=3|year=1956|pages=239–244|doi=10.1007/BF02589410|doi-access=free}}.
• {{cite book

|author=Beauzamy, Bernard

|title=Introduction to Banach Spaces and their Geometry

|year=1985 |orig-year=1982

|edition=Second revised

|publisher=North-Holland

|isbn=0-444-86416-4

}}

• {{cite journal

|doi=10.1007/BF02762802 | doi-access=

|author=Per Enflo

|author-link=Per Enflo

|title=Banach spaces which can be given an equivalent uniformly convex norm

|journal=Israel Journal of Mathematics

|volume=13

|issue=3–4

|year=1972

|pages=281–288}}

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

{{Functional analysis}}

Category:Convex analysis

Category:Banach spaces