Abstract
Letf be a real-valued function defined on the product ofm finite-dimensional open convex setsX 1, ⋯,X m .
Assume thatf is quasiconvex and is the sum of nonconstant functionsf 1, ⋯,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 off 1 andf 2, or, equivalently, in terms of the separation of the graphs off 1 andf 2 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.
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Debreu, G., Koopmans, T.C. Additively decomposed quasiconvex functions. Mathematical Programming 24, 1–38 (1982). https://doi.org/10.1007/BF01585092
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DOI: https://doi.org/10.1007/BF01585092