Modulus and characteristic of convexity: Difference between revisions

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In [[mathematics]], the '''modulus and characteristic of convexity''' are measures of "how [[convex set|convex]]" the [[unit ball]] in a [[Banach space]] is. In some sense, the modulus of convexity has the same relationship to the ''εε''-''δδ'' definition of [[uniformly convex space|uniform convexity]] as the [[modulus of continuity]] does to the ''εε''-''δδ'' definition of [[continuous function|continuity]].

==Definitions==

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

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

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

:<math>\varepsilon_{0} = \sup \{ \varepsilon | \delta(\varepsilon) = 0 \}.</math>

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==Properties==

* The modulus of convexity, ''&delta;δ''(''&epsilon;ε''), is a [[monotonic function|non-decreasing]] and [[convex function]] of ''&epsilon;ε''.

* (''X'',&nbsp;||&nbsp;||) is a [[uniformly convex space]] [[if and only if]] its characteristic of convexity ''&epsilon;ε''₀&nbsp;=&nbsp;0.

* (''X'',&nbsp;||&nbsp;||) is a [[strictly convex space]] (i.e., the boundary of the unit ball ''B'' contains no line segments) if and only if ''&delta;δ''(2)&nbsp;=&nbsp;1.

==ReferenceReferences==

* {{cite journal