A Rockafellar Theorem for cyclically quasi-monotone maps: the regular non-vanishing case

TL;DR

This paper extends Rockafellar's theorem to cyclically quasi-monotone maps, demonstrating that under certain regularity conditions, such maps are included in the normal cone of a quasi-convex function, with implications for economics and optimal transport.

Contribution

It proves a positive result for $ abla$-regular, non-vanishing maps and discusses the connection to economic revealed preferences and $L^ abla$ optimal transport, providing explicit examples.

Findings

Positive inclusion result for $ abla$-regular, non-vanishing maps

Connections established to economic revealed preference theory

Examples illustrating main challenges in the general case

Abstract

We study the connection between cyclic quasi-monotonicity and quasi-convexity, focusing on whether every cyclically quasi-monotone (possibly multivalued) map is included in the normal cone operator of a quasi-convex function, in analogy with Rockafellar's theorem for convex functions. We provide a positive answer for $\mathscr{C}^1$-regular, non-vanishing maps in any dimension, as well as for general multi-maps in dimension $1$. We further discuss connections to revealed preference theory in economics and to $L^\infty$ optimal transport. Finally, we present explicit constructions and examples, highlighting the main challenges that arise in the general case.