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In mathematics, the modulus of convexity and the characteristic of convexity are measures of "how convex" the unit ball in a Banach space is. In some sense, the modulus of convexity has the same relationship to the ''ε''-''δ'' definition of uniform convexity as the modulus of continuity does to the ''ε''-''δ'' definition of continuity. ==Definitions== The modulus of convexity of a Banach space (''X'', || ||) is the function defined by : where ''S'' denotes the unit sphere of (''X'', || ||). In the definition of ''δ''(''ε''), one can as well take the infimum over all vectors ''x'', ''y'' in ''X'' such that and .〔p. 60 in .〕 The characteristic of convexity of the space (''X'', || ||) is the number ''ε''0 defined by : These notions are implicit in the general study of uniform convexity by J. A. Clarkson (; 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Modulus and characteristic of convexity」の詳細全文を読む スポンサード リンク
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