Additively decomposed quasiconvex functions

Abstract

Letf be a real-valued function defined on the product ofm finite-dimensional open convex setsX1, ź,Xm. Assume thatf is quasiconvex and is the sum of nonconstant functionsf1, ź,fm defined on the respective factor sets. Then everyfi is continuous; with at most one exception every functionfi is convex; if the exception arises, all the other functions have a strict convexity property and the nonconvex function has several of the differentiability properties of a convex function. We define the convexity index of a functionfi appearing as a term in an additive decomposition of a quasiconvex function, and we study the properties of that index. In particular, in the case of two one-dimensional factor sets, we characterize the quasiconvexity of an additively decomposed functionf either in terms of the nonnegativity of the sum of the convexity indices off1 andf2, or, equivalently, in terms of the separation of the graphs off1 andf2 by means of a logarithmic function. We investigate the extension of these results to the case ofm factor sets of arbitrary finite dimensions. The introduction discusses applications to economic theory.