Modulus and characteristic of convexity: Difference between revisions

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* When ''X'' is uniformly convex, it admits an equivalent norm with power type modulus of convexity.<ref>see {{citation

| last=Pisier |first=Gilles |authorlink=Gilles Pisier

| title= Martingales with values in uniformly convex spaces | journal=Israel J. Math. | volume=20 | year=1975 | issue=3–4 | pages=326–350 | doi = 10.1007/BF02760337 | url=http://www.springerlink.com/content/pwh1126545520581/ | mr=394135}}

.</ref> Namely, there exists {{nowrap|''q'' &ge; 2}} and a constant&nbsp;{{nowrap|''c'' &gt; 0}} such that

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

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| volume = 3

| year = 1955

| pages = 239-244239–244

| doi = 10.1007/BF02589410

}}</ref> If <math>1<p\le2</math>, then it satisfies the following implicit equation: