Quadratic Cost Equivalence with Classical Cyclical Monotonicity is a result from the foundations of optimal transport. For mathematical objects, let mathematical objects, and define the quadratic cost function c: d d & (x,y) & 2|x-y|2. Then mathematical objects is mathematical objects-cyclically monotone, in the sense that for . It helps organize the relationship between Monge maps, Kantorovich plans, duality, and Wasserstein geometry.