Additively decomposed quasiconvex functions

Abstract

In a recently published paper with the same title, Debreu and Koopmans have studied conditions which imply the quasiconvexity of the function

$$f(x) = \mathop \sum \limits_{i = 1}^n f_i (x_i ),x_i \in X_i ,$$

wherex = (x ₁,x ₂,⋯,x _n ) and, fori = 1, 2,⋯, n,X ₁ is a finite-dimensional open convex set andf _i a real-valued nonconstant function onX ₁ These conditions involve the convexity indices of functionsf _i, a concept introduced in the Debreu and Koopmans paper. First, we give a new definition of the convexity index equivalent to that of Debreu and Koopmans. Then, by means of this definition, we can simplify the proofs given by Debreu and Koopmans and extend some of their results.