Brenier theorem
= Brenier theorem
{c}
= Brenier's theorem
{c}
{synonym}
For <probability measures> on $\mathbb R^d$ with finite second <moments> and a source satisfying <absolute continuity of measures> with respect to <Lebesgue measure>, the quadratic <Kantorovich optimal transport problem> has a unique optimal <transport plan>. It is induced by the <gradient> of a <convex function>, which also uniquely solves the <Monge optimal transport problem> up to a source-null set.