Modulus and characteristic of convexity: Difference between revisions

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* The Banach space {{nowrap|(''X'', ǁ ⋅ ǁ)}} 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 {{citation

| last=Pisier |first=Gilles |authorlinkauthor-link=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 | mr=394135|s2cid=120947324 }}

.</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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==References==

* {{cite book|author=Beauzamy, Bernard|title=Introduction to Banach Spaces and their Geometry|year=1985 |origyearorig-year=1982|edition=Second revised|publisher=North-Holland|mr=889253|isbn=0-444-86416-4}}

*{{citation

| last = Clarkson