Additively decomposed quasiconvex functions

Letf be a real-valued function defined on the product ofm finite-dimensional open convex setsX₁, ,X_m.Assume thatf is quasiconvex and is the sum of nonconstant functionsf₁, ,f_m defined on the respective factor sets. Then everyf_i is continuous; with at most one exception every functionf_i 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 functionf_i 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.