Modulus and characteristic of convexity

In mathematics, the modulus of convexity and the characteristic of convexity are measures of "how convex" the unit ball in a Banach space is. In some sense, the modulus of convexity has the same relationship to the ε-δ definition of uniform convexity as the modulus of continuity does to the ε-δ definition of continuity.

Definitions

The modulus of convexity of a Banach space (X, || ||) is the function δ : [0, 2] → [0, 1] defined by

\delta (\varepsilon) = \inf \left\{ 1 - \left\| \frac{x + y}{2} \right\| \,:\,  x, y \in S, \| x - y \| \geq \varepsilon \right\},

where S denotes the unit sphere of (X, || ||). In the definition of δ(ε), one can as well take the infimum over all vectors x, y in X such that ǁxǁ, ǁyǁ 1 and ǁx yǁ ε.[1]

The characteristic of convexity of the space (X, || ||) is the number ε0 defined by

\varepsilon_{0} = \sup \{ \varepsilon \,:\, \delta(\varepsilon) = 0 \}.

These notions are implicit in the general study of uniform convexity by J. A. Clarkson (Clarkson (1936); this is the same paper containing the statements of Clarkson's inequalities). The term "modulus of convexity" appears to be due to M. M. Day.[2]

Properties

\delta(\varepsilon / 2) \le \delta_1(\varepsilon) \le \delta(\varepsilon), \quad \varepsilon \in [0, 2].
\delta(\varepsilon) \ge c \, \varepsilon^q, \quad \varepsilon \in [0, 2].

See also

Notes

  1. p. 60 in Lindenstrauss & Tzafriri (1979).
  2. Day, Mahlon (1944), "Uniform convexity in factor and conjugate spaces", Ann. of Math. (2) (Annals of Mathematics) 45 (2): 375385, doi:10.2307/1969275, JSTOR 1969275
  3. Lemma 1.e.8, p. 66 in Lindenstrauss & Tzafriri (1979).
  4. see Remarks, p. 67 in Lindenstrauss & Tzafriri (1979).
  5. see Proposition 1.e.6, p. 65 and Lemma 1.e.7, 1.e.8, p. 66 in Lindenstrauss & Tzafriri (1979).
  6. see Pisier, Gilles (1975), "Martingales with values in uniformly convex spaces", Israel J. Math. 20 (3–4): 326–350, doi:10.1007/BF02760337, MR 394135 .

References

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