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.
The modulus of convexity of a Banach space (X, || ||) is the function δ : [0, 2] → [0, 1] defined by
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.
The characteristic of convexity of the space (X, || ||) is the number ε0 defined by
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.
- 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]. The modulus of convexity need not itself be a convex function of ε. However, the modulus of convexity is equivalent to a convex function in the following sense: there exists a convex function δ1(ε) such that
- 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. Namely, there exists q ≥ 2 and a constant c > 0 such that
- p. 60 in Lindenstrauss & Tzafriri (1979).
- 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
- Lemma 1.e.8, p. 66 in Lindenstrauss & Tzafriri (1979).
- see Remarks, p. 67 in Lindenstrauss & Tzafriri (1979).
- see Proposition 1.e.6, p. 65 and Lemma 1.e.7, 1.e.8, p. 66 in Lindenstrauss & Tzafriri (1979).
- 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 .
- Beauzamy, Bernard (1985) . 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. MR1904276
- 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.