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 is the function defined by
where S denotes the unit sphere of. In the definition of δ, one can as well take the infimum over all vectors x, y in X such that and.
The characteristic of convexity of the space is the number ε0 defined by
These notions are implicit in the general study of uniform convexity by J. A. Clarkson. The term "modulus of convexity" appears to be due to M. M. Day.

Properties

The modulus of convexity is known for the L^p spaces. If, then it satisfies the following implicit equation:
Knowing that one can suppose that. Substituting this into the above, and expanding the left-hand-side as a Taylor series around, one can calculate the coefficients:
For, one has the explicit expression
Therefore,.