Modulus and characteristic of convexity: Difference between revisions

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==Properties==

* The modulus of convexity, ''δ''(''ε''), is a [[monotonic function|non-decreasing]] function of ''ε'', and {{nowrap|''δ''(''ε'')&thinsp;/&thinsp;''ε''}} is also non-decreasing on&nbsp;{{nowrap|(0, 2]}}.<ref>Lemma 1.e.8, p&nbsp;66 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.</ref> The modulus of convexity need not itself be a [[convex function]] of&nbsp;''ε''.<ref>see Remarks, p.&nbsp;67 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.</ref>

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

* The modulus of convexity, ''δ''(''ε''), is a [[monotonic function|non-decreasing]] function of ''ε''. (The modulus of convexity need not itself be a [[convex function]] of ''ε''.<ref>p. 67 in [[Lindenstrauss, Joram]]; Tzafriri, Lior, "Classical Banach spaces. II. Function spaces". ''Ergebnisse der Mathematik und ihrer Grenzgebiete'' [Results in Mathematics and Related Areas], 97. ''Springer-Verlag, Berlin-New York,'' 1979. x+243 pp.</ref>)

* (''X'',&nbsp;||&nbsp;||) is a [[uniformly convex space]] [[if and only if]] its characteristic of convexity ''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 ''δ''(2)&nbsp;=&nbsp;1.