Modulus and characteristic of convexity: Difference between revisions

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==Definitions==

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

:<math>\delta (\varepsilon) = \inf \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 In the definition of&nbsp;''δ'characteristic of convexity'(''ε''), ofone can as well take the spaceinfimum over all vectors (''Xx'', ''y'' in&nbsp;''X'' such that {{nowrap||ǁ''x''ǁ, ǁ''y''ǁ &nbsple; 1}} and {{nowrap||)ǁ''x'' is&minus; the''y''ǁ number&ge; ''ε''}}.<subref>0p.&nbsp;60 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.</subref> defined by

The '''characteristic of convexity''' of the space (''X'',&nbsp;||&nbsp;||) is the number ''ε''₀ defined by

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

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

These notions are implicit in the general study of uniform convexity by J. A. Clarkson (see below; this is the same paper containing the statements of [[Clarkson's inequalities]]). The term "modulus of convexity" appears to be due to M. M. Day (see reference below).

These notions are implicit in the general study of uniform convexity by J. &nbsp;A. &nbsp;Clarkson (see below{{harvtxt|Clarkson|1936}}; this is the same paper containing the statements of [[Clarkson's inequalities]]). The term "modulus of convexity" appears to be due to M. &nbsp;M. &nbsp;Day (see reference below{{harvtxt|Day|1944}}).

==Properties==