Modulus and characteristic of convexity#Definitions

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

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

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 {{nowrap|ǁxǁ, ǁyǁ ≤ 1}} and {{nowrap|ǁx − yǁ ≥ ε}}.p. 60 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}.

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

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

These notions are implicit in the general study of uniform convexity by J. A. Clarkson ({{harvtxt|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.{{citation

| last = Day

| first = Mahlon

| title = Uniform convexity in factor and conjugate spaces

| journal = Annals of Mathematics |series = 2

| volume = 45

| year = 1944

| pages = 375–385

| doi = 10.2307/1969275

| issue = 2

| jstor = 1969275

}}

• The modulus of convexity, δ(ε), is a non-decreasing function of ε, and the quotient {{nowrap|δ(ε) / ε}} is also non-decreasing on {{nowrap|(0, 2]}}.Lemma 1.e.8, p. 66 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}. The modulus of convexity need not itself be a convex function of ε.see Remarks, p. 67 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}. However, the modulus of convexity is equivalent to a convex function in the following sense:see Proposition 1.e.6, p. 65 and Lemma 1.e.7, 1.e.8, p. 66 in {{harvtxt|Lindenstrauss|Tzafriri|1979}}. there exists a convex function δ₁(ε) such that

::$\backslash delta(\backslash varepsilon\; /\; 2)\; \backslash le\; \backslash delta\_1(\backslash varepsilon)\; \backslash le\; \backslash delta(\backslash varepsilon),\; \backslash quad\; \backslash varepsilon\; \backslash in\; [0,\; 2].$

• The normed space {{nowrap|(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 {{nowrap|δ(ε) > 0}} for every {{nowrap|ε > 0}}.
• The Banach space {{nowrap|(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.see {{citation

| last=Pisier |first=Gilles |author-link=Gilles Pisier

| title= Martingales with values in uniformly convex spaces | journal=Israel Journal of Mathematics | volume=20 | year=1975 | issue=3–4 | pages=326–350 | doi = 10.1007/BF02760337 | doi-access= | mr=394135|s2cid=120947324 }}

. Namely, there exists {{nowrap|q ≥ 2}} and a constant {{nowrap|c > 0}} such that

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

The modulus of convexity is known for the L^P spaces.{{citation

| last = Hanner

| first = Olof

| title = On the uniform convexity of $L^p$ and $\backslash ell^p$

| journal = Arkiv för Matematik

| volume = 3

| year = 1955

| pages = 239–244

| doi = 10.1007/BF02589410

| doi-access = free

}} If

:$\backslash left(1-\backslash delta\_p(\backslash varepsilon)+\backslash frac\{\backslash varepsilon\}\{2\}\backslash right)^p+\backslash left(1-\backslash delta\_p(\backslash varepsilon)-\backslash frac\{\backslash varepsilon\}\{2\}\backslash right)^p=2.$

Knowing that $\backslash delta\_p(\backslash varepsilon+)=0,$ one can suppose that $\backslash delta\_p(\backslash varepsilon)=a\_0\backslash varepsilon+a\_1\backslash varepsilon^2+\backslash cdots$. Substituting this into the above, and expanding the left-hand-side as a Taylor series around $\backslash varepsilon=0$, one can calculate the $a\_i$ coefficients:

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

For

:$\backslash delta\_p(\backslash varepsilon)=1-\backslash left(1-\backslash left(\backslash frac\{\backslash varepsilon\}\{2\}\backslash right)^p\backslash right)^\{\backslash frac1p\}.$

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

• Uniformly smooth space

{{reflist}}

• {{cite book|author=Beauzamy, Bernard|title=Introduction to Banach Spaces and their Geometry|year=1985 |orig-year=1982|edition=Second revised|publisher=North-Holland|mr=889253|isbn=0-444-86416-4}}
• {{citation

| last = Clarkson

| first = James

| title = Uniformly convex spaces

| journal = Transactions of the American Mathematical Society

| volume = 40

| year = 1936

| pages = 396–414

| doi = 10.2307/1989630

| issue = 3

| publisher = American Mathematical Society

| jstor = 1989630

| doi-access = free

}}

• 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.
• {{citation

| last1 = Lindenstrauss

| first1 = Joram | author1-link = Joram Lindenstrauss

| last2 = Tzafriri | first2 = Lior

| title = Classical Banach spaces. II. Function spaces

| series = Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas]

| volume = 97

| publisher = Springer-Verlag

| location = Berlin-New York

| year = 1979

| pages = 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.

{{Banach spaces}}

{{Functional analysis}}

Category:Banach spaces

Category:Convex analysis