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

• The modulus of convexity, δ(ε), is a non-decreasing function of ε, and the quotient δ(ε) / ε is also non-decreasing on (0, 2].^[3] The modulus of convexity need not itself be a convex function of ε.^[4] However, the modulus of convexity is equivalent to a convex function in the following sense:^[5] there exists a convex function δ₁(ε) such that

$\delta(\varepsilon / 2) \le \delta_1(\varepsilon) \le \delta(\varepsilon), \quad \varepsilon \in [0, 2].$

• The normed space (X, ǁ ⋅ ǁ) is uniformly convex if and only if its characteristic of convexity ε₀ is equal to 0, i.e., if and only if δ(ε) > 0 for every ε > 0.
• The Banach space (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 points (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.^[6] Namely, there exists q ≥ 2 and a constant c > 0 such that

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

See also

• Uniformly smooth space

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): 375–385, 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

• Beauzamy, Bernard (1985) [1982]. Introduction to Banach Spaces and their Geometry (Second revised ed.). North-Holland. ISBN 0-444-86416-4. MR 889253.
• Clarkson, James (1936), "Uniformly convex spaces", Trans. Amer. Math. Soc. (American Mathematical Society) 40 (3): 396–414, doi:10.2307/1989630, JSTOR 1989630
• Fuster, Enrique Llorens. Some moduli and constants related to metric fixed point theory. Handbook of metric fixed point theory, 133-175, Kluwer Acad. Publ., Dordrecht, 2001. MR 1904276
• Lindenstrauss, Joram and Benyamini, Yoav. Geometric nonlinear functional analysis Colloquium publications, 48. American Mathematical Society.
• Lindenstrauss, Joram; Tzafriri, Lior (1979), Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas] 97, Berlin-New York: Springer-Verlag, pp. x+243, ISBN 3-540-08888-1.
• Vitali D. Milman. Geometric theory of Banach spaces II. Geometry of the unit sphere. Uspechi Mat. Nauk, vol. 26, no. 6, 73-149, 1971; Russian Math. Surveys, v. 26 6, 80-159.