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→Properties: there is a convex function equivalent to the modulus |
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==Properties==
* The modulus of convexity, ''δ''(''ε''), is a [[monotonic function|non-decreasing]] function of ''ε'', and the quotient {{nowrap|''δ''(''ε'') / ''ε''}} is also non-decreasing on {{nowrap|(0, 2]}}.<ref>Lemma 1.e.8, p. 66 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.</ref> The modulus of convexity need not itself be a [[convex function]] of ''ε''.<ref>see Remarks, p. 67 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.</ref> However, the modulus of convexity is equivalent to a convex function in the following sense:<ref>see Proposition 1.e.6, p. 65 and Lemma 1.e.7, 1.e.8, p. 66 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.</ref> there exists a convex function ''δ''<sub>1</sub>(''ε'') such that
::<math>\delta(\varepsilon / 2) \le \delta_1(\varepsilon) \le \delta(\varepsilon), \quad \varepsilon \in [0, 2].</math>
* (''X'', || ||) is a [[uniformly convex space]] [[if and only if]] its characteristic of convexity ''ε''<sub>0</sub> is equal to 0, ''i.e.'', if and only if {{nowrap|''δ''(''ε'') > 0}} for every {{nowrap|''ε'' > 0}}.▼
▲* The normed space (''X'', || ||) is
* (''X'', || ||) is a [[strictly convex space]] (i.e., the boundary of the unit ball ''B'' contains no line segments) if and only if ''δ''(2) = 1, ''i.e.'', if only [[antipodal point]]s (of the form ''x'' and ''y'' = −''x'') of the unit sphere can have distance equal to 2.
* When ''X'' is uniformly convex, it admits an equivalent norm with power type modulus of convexity.<ref>see {{harvtxt|Pisier|1975}}.</ref> Namely, there exists {{nowrap|''q'' ≥ 2}} and a constant {{nowrap|''c'' > 0}} such that
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