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* (''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}}.
* (''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
::<math>\delta(\varepsilon) \ge c \, \varepsilon^q, \quad \varepsilon \in [0, 2].</math>
==Notes==
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