Logconcave and Quasi-Concave Measures

Abstract

A nonnegative function f defined on a convex subset A of the space R ^m is said to be logarithmically concave (logconcave) if for every pair x, y ∈ A and 0 < λ < 1 we have the inequality

$$ f\left( {\lambda x + \left( {1 - \lambda } \right)y} \right) \geqslant {\left[ {f\left( x \right)} \right]^\lambda }{\left[ {f\left( y \right)} \right]^{1 - \lambda }} $$

(4.1.1)

If f is positive valued, then log f is a concave function on A. If the inequality in (4.1.1) is reversed, then f is said to be logarithmically convex (logconvex) on the set A. If the inequality holds strictly for x ≠ y, then f is said to be strictly logconcave (logconvex).