OPTIMIZATION FOR PRODUCTS OF CONCAVE FUNCTIONS

If h denotes the product of finitely many concave non-negative functions on a compact interval [a, b], then it is shown that there exist points alpha and beta with a <= alpha <= beta <= b such that h is strictly increasing on [alpha, alpha), constant on (alpha, beta), and strictly decreasing on (beta, b]. This structure theorem leads to an extension of several classical optimization results for concave functions on convex sets to the case of products of concave functions.