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 convex space|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 bywhere 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, δ, is a non-decreasing function of ε, and the quotient is also non-decreasing on . 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 such that
- The normed space is uniformly convex if and only if its characteristic of convexity ε0 is equal to 0, i.e., if and only if for every .
- The Banach space is a strictly convex space if and only if δ = 1, i.e., if only antipodal points 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. Namely, there exists and a constant such that
Modulus of convexity of the ''L''''P'' spaces
The modulus of convexity is known for the LP 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,.