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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

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

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

$${\displaystyle \varepsilon _{0}=\sup\{\varepsilon \,:\,\delta (\varepsilon )=0\}.}$$

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

$${\displaystyle \delta (\varepsilon /2)\leq \delta _{1}(\varepsilon )\leq \delta (\varepsilon ),\quad \varepsilon \in [0,2].}$$

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

$${\displaystyle \delta (\varepsilon )\geq c\,\varepsilon ^{q},\quad \varepsilon \in [0,2].}$$

Modulus of convexity of the L p {\displaystyle L^{p}} L^{p} spaces

The modulus of convexity is known for the L^p spaces.[7] If $${\displaystyle 1<p\leq 2}$$, then it satisfies the following implicit equation:

$${\displaystyle \left(1-\delta _{p}(\varepsilon )+{\frac {\varepsilon }{2}}\right)^{p}+\left(1-\delta _{p}(\varepsilon )-{\frac {\varepsilon }{2}}\right)^{p}=2.}$$

Knowing that $${\displaystyle \delta _{p}(\varepsilon +)=0,}$$ one can suppose that $${\displaystyle \delta _{p}(\varepsilon )=a_{0}\varepsilon +a_{1}\varepsilon ^{2}+\cdots }.$$ Substituting this into the above, and expanding the left-hand-side as a Taylor series around $$\varepsilon=0,$$ one can calculate the $$a_{i}$$ coefficients:

$${\displaystyle \delta _{p}(\varepsilon )={\frac {p-1}{8}}\varepsilon ^{2}+{\frac {1}{384}}(3-10p+9p^{2}-2p^{3})\varepsilon ^{4}+\cdots .}$$

For $${\displaystyle 2<p<\infty }, one has the explicit expression \( {\displaystyle \delta _{p}(\varepsilon )=1-\left(1-\left({\frac {\varepsilon }{2}}\right)^{p}\right)^{\frac {1}{p}}.}$$

Therefore, $${\displaystyle \delta _{p}(\varepsilon )={\frac {1}{p2^{p}}}\varepsilon ^{p}+\cdots }.$$

Uniformly smooth space

Notes

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 0394135 .

Hanner, Olof (1955), "On the uniform convexity of $$L^{p}$$ and $$\ell ^{p}$$", Arkiv för Mathematik, 3: 239–244, doi:10.1007/BF02589410

References

Beauzamy, Bernard (1985) [1982]. Introduction to Banach Spaces and their Geometry (Second revised ed.). North-Holland. ISBN 0-444-86416-4. MR 0889253.
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.

vte

Functional analysis (topics – glossary)
Spaces

Hilbert space Banach space Fréchet space topological vector space

Theorems

Hahn–Banach theorem closed graph theorem uniform boundedness principle Kakutani fixed-point theorem Krein–Milman theorem min-max theorem Gelfand–Naimark theorem Banach–Alaoglu theorem

Operators

bounded operator compact operator adjoint operator unitary operator Hilbert–Schmidt operator trace class unbounded operator

Algebras

Banach algebra C*-algebra spectrum of a C*-algebra operator algebra group algebra of a locally compact group von Neumann algebra

Open problems

invariant subspace problem Mahler's conjecture

Applications

Besov space Hardy space spectral theory of ordinary differential equations heat kernel index theorem calculus of variation functional calculus integral operator Jones polynomial topological quantum field theory noncommutative geometry Riemann hypothesis

locally convex space approximation property balanced set Schwartz space weak topology barrelled space Banach–Mazur distance Tomita–Takesaki theory