Modulus and characteristic of convexity

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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

δ ( ε ) = inf { 1 − ‖ x + y 2 ‖ : x , y ∈ S , ‖ x − y ‖ ≥ ε } , {\displaystyle \delta (\varepsilon )=\inf \left\{1-\left\|{\frac {x+y}{2}}\right\|\,:\,x,y\in S,\|x-y\|\geq \varepsilon \right\},} {\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

ε 0 = sup { ε : δ ( ε ) = 0 } . {\displaystyle \varepsilon _{0}=\sup\{\varepsilon \,:\,\delta (\varepsilon )=0\}.} {\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
δ ( ε / 2 ) ≤ δ 1 ( ε ) ≤ δ ( ε ) , ε ∈ [ 0 , 2 ] . {\displaystyle \delta (\varepsilon /2)\leq \delta _{1}(\varepsilon )\leq \delta (\varepsilon ),\quad \varepsilon \in [0,2].} {\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 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
δ ( ε ) ≥ c ε q , ε ∈ [ 0 , 2 ] . {\displaystyle \delta (\varepsilon )\geq c\,\varepsilon ^{q},\quad \varepsilon \in [0,2].} {\displaystyle \delta (\varepsilon )\geq c\,\varepsilon ^{q},\quad \varepsilon \in [0,2].}

Modulus of convexity of the LP spaces

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

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

Knowing that δ p ( ε + ) = 0 , {\displaystyle \delta _{p}(\varepsilon +)=0,} {\displaystyle \delta _{p}(\varepsilon +)=0,} one can suppose that δ p ( ε ) = a 0 ε + a 1 ε 2 + ⋯ {\displaystyle \delta _{p}(\varepsilon )=a_{0}\varepsilon +a_{1}\varepsilon ^{2}+\cdots } {\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 ε = 0 {\displaystyle \varepsilon =0} {\displaystyle \varepsilon =0}, one can calculate the a i {\displaystyle a_{i}} {\displaystyle a_{i}} coefficients:

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

For 2 < p < ∞ {\displaystyle 2<p<\infty } {\displaystyle 2<p<\infty }, one has the explicit expression

δ p ( ε ) = 1 − ( 1 − ( ε 2 ) p ) 1 p . {\displaystyle \delta _{p}(\varepsilon )=1-\left(1-\left({\frac {\varepsilon }{2}}\right)^{p}\right)^{\frac {1}{p}}.} {\displaystyle \delta _{p}(\varepsilon )=1-\left(1-\left({\frac {\varepsilon }{2}}\right)^{p}\right)^{\frac {1}{p}}.}

Therefore, δ p ( ε ) = 1 p 2 p ε p + ⋯ {\displaystyle \delta _{p}(\varepsilon )={\frac {1}{p2^{p}}}\varepsilon ^{p}+\cdots } {\displaystyle \delta _{p}(\varepsilon )={\frac {1}{p2^{p}}}\varepsilon ^{p}+\cdots }.

See also

Notes

  1. p. 60 in Lindenstrauss & Tzafriri (1979).
  2. Day, Mahlon (1944), "Uniform convexity in factor and conjugate spaces", Annals of Mathematics, 2, 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 Journal of Mathematics, 20 (3–4): 326–350, doi:10.1007/BF02760337, MR 0394135, S2CID 120947324 .
  7. Hanner, Olof (1955), "On the uniform convexity of L p {\displaystyle L^{p}} {\displaystyle L^{p}} and ℓ p {\displaystyle \ell ^{p}} {\displaystyle \ell ^{p}}", Arkiv för Matematik, 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", Transactions of the American Mathematical Society, 40 (3), American Mathematical Society: 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], vol. 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.