Uniformly convex space

Properties

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

• 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 ${\displaystyle \{f_n{\}}_{n=1}^{\mathrm{\infty }}}$ is a sequence in a uniformly convex Banach space that converges weakly to ${\displaystyle f}$ and satisfies ${\displaystyle \Vert f_n\Vert \to \Vert f\Vert ,}$ then ${\displaystyle f_n}$ converges strongly to ${\displaystyle f}$, that is, ${\displaystyle \Vert f_n-f\Vert \to 0}$.
• A Banach space ${\displaystyle X}$ is uniformly convex if and only if its dual ${\displaystyle X^{\ast }}$ is uniformly smooth.
• Every uniformly convex space is strictly convex. Intuitively, the strict convexity means a stronger triangle inequality ${\displaystyle \Vert x+y\Vert <\Vert x\Vert +\Vert y\Vert }$ whenever ${\displaystyle x,y}$ are linearly independent, while the uniform convexity requires this inequality to be true uniformly.

Examples

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