Modulus and characteristic of convexity: Difference between revisions

Revision as of 00:00, 12 November 2012

Revision as of 00:01, 12 November 2012

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The '''modulus of convexity''' of a Banach space (''X'', || ||) is the function ''δ'' : [0, 2] → [0, 1] defined by

:<math>\delta (\varepsilon) = \supinf \left\{ \left. 1 - \left\| \frac{x + y}{2} \right\| \, \right| x, y \in S, \| x - y \| \geq \varepsilon \right\},</math>

where ''S'' denotes the unit sphere of (''X'',&nbsp;||&nbsp;||). The '''characteristic of convexity''' of the space (''X'',&nbsp;||&nbsp;||) is the number ''ε''₀ defined by