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

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\},}

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

ε

0

=

sup

{

ε

:

δ

(

ε

)

=

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.

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]. 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

δ

(

ε

/

2

)

≤

δ

1

(

ε

)

≤

δ

(

ε

)

,

ε

∈

[

0

,

2

]

.