Mathematical Research Letters
Volume 21 (2014)
Obstructions to regularity in the classical Monge problem
Pages: 697 – 712
We provide counter-examples to regularity of optimal maps in the classical Monge problem under various assumptions on the initial data. Our construction is based on a variant of the counter-example in  to Lipschitz regularity of the monotone optimal map between smooth densities supported on convex domains.