preprint

Inserted: 15 oct 2026
Last Updated: 15 oct 2026

Year: 2026

Abstract:

Rockafellar showed that every cyclically monotone operator is contained in the subdifferential of a convex function, and characterized the maximal ones. We establish the quasi-convex analogue of this theorem, with no assumption on the operator: in any separable Banach space, a setting we show to be sharp, every cyclically quasi-monotone operator is contained in the normal cone operator of a quasi-convex function. In finite dimension, we characterize maximal cyclically quasi-monotone operators and show that they are maximal quasi-monotone. Unlike in the convex case, the potential is not unique, and we build it from a preference relation obtained through successive preorders. As applications, we extend Afriat's theorem on revealed preferences in economics to infinite data sets, and obtain Kantorovich-type potentials for a supremal optimal transport problem.

Keywords: cyclical monotonicity, $L^\infty$ optimal transport, quasi-convex functions, cyclical quasi-monotonicity, quasi-monotone operators, revelead preference, generalized convexity, Afriat's theorem, normal cone, utility representation