Partial regularity of Brenier solutionsof the Monge-Ampère equation

Given Ω,Λ ⊂ ⁿ two bounded open sets, andf and g two probability densities concentrated on Ω and Λrespectively, we investigate the regularity of the optimal map∇ φ (the optimality referring to the Euclidean quadratic cost) sending f onto g. We show that if f andg are both bounded away from zero and infinity, we can findtwo open sets Ω'⊂ Ω and Λ'⊂Λ such that f and g are concentrated on Ω' andΛ' respectively, and ∇φ:Ω' → Λ'is a (bi-Hölder) homeomorphism. This generalizes the $2$-dimensional partial regularityresult of [8].