Optimal Transport: Some Analytic and Geometric Aspects

Abstract

Let µ and ν be probability measures on R n , and let c : R n ×R n → [0, ∞) be a cost function. In its basic form, the optimal transport problem is to find a map T : R n → R n pushing the measure µ forward to ν, minimizing the cost of transportation R Rn c(x, T(x)) dµ(x). We review the reformulation of this problem in the context of the Kantorovich duality and describe some results on optimal transport with the quadratic cost function c(x, y) = |x−y| 2/2, in particular Brenier’s polar factorization theorem and its consequences. Finally, we discuss some results on the regularity of optimal maps in the quadratic cost setting and give a new condition guaranteeing the discontinuity of optimal maps in R n .