A sharp inequality for transport maps in W1,p(R)W1,p(R) via approximation

For ff convex and increasing, we prove the inequality ∫f(|U′|)≥∫f(nT′)∫f(|U′|)≥∫f(nT′), every time that UU is a Sobolev function of one variable and TT is the non-decreasing map defined on the same interval with the same image measure as UU, and the function n(x)n(x) takes into account the number of pre-images of UU at each point. This may be applied to some variational problems in a mass-transport framework or under volume constraints.