Optimal Mass Transport and Solution to the Far Field Refractor Problem

1. Optimal transportation set up Let D, D∗ be two domains on Sn−1 or bounded domains in a manifold (D might be contained in one manifold and D∗ in another) with |∂D| = 0. LetN be a multi-valued mapping from D onto D∗ such thatN(x) is single-valued a.e. on D. For F ⊂ D∗, we set T (F) = N−1(F) = {x ∈ D : N(x) ∩ F , ∅}. We sayN is measurable if T (F) is Lebesgue measurable for any Borel set F ⊂ D∗. For example, ifN = ∂u with u convex, thenN is measurable (see exercises). Given nonnegative g ∈ L1(D) and a finite Radon measure Γ on D∗ satisfying ∫ D g(x) dx = Γ(D∗) > 0, we sayN is measure preserving from g(x)dx to Γ if for any Borel F ⊂ D∗