Consider the class of probability distributions on the unit square with uniform marginal distributions. The class of associated regression functions is characterized by integral inequalities or, equivalently, as forming the closed convex hull of 1-1, measure-preserving mappings of the unit interval onto itself.
"Conditions on the Regression Function when Both Variables are Uniformly Distributed." Ann. Probab. 4 (5) 869 - 873, October, 1976. https://doi.org/10.1214/aop/1176995992