Uniformly convex space

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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.


A uniformly convex space is a normed vector space so that, for every 0<\epsilon \leq 2 there is some \delta>0 so that for any two vectors with \|x\| = 1 and \|y\| = 1, the condition

\left\|\frac{x+y}{2}\right\| \geq 1-\delta.

implies that:

\|x-y\|\leq \epsilon

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


  • The Milman–Pettis theorem states that every uniformly convex Banach space is reflexive, while the converse is not true.
  • If  \{f_n\}_{n=1}^{\infty} is a sequence in a uniformly convex Banach space which 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.


  • Every Hilbert space is uniformly convex.
  • Every closed subspace of a uniformly convex Banach space is uniformly convex.
  • Hanner's inequalities imply that Lp spaces (1<p<\infty) are uniformly convex.
  • Conversely, L^\infty is not uniformly convex. For example, in \mathbb{R}^2 consider x=(1,1) and y=(0,1). Then \|x\|_{\infty}=\|y\|_{\infty}=1 and \|x+y\|_{\infty}=\|(1,2)\|_{\infty}=2, but \|x-y\|_{\infty}=\|(1,0)\|_{\infty}=1.

See also[edit]


  • 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.