Optimal Transport and the ABP Method in Higher Codimension

Abstract: We prove an inverse optimal-transport principle for the ABP contact set of animmersed submanifold. A fixed function on the submanifold determines, for every probability density on a bounded convex target, a source measure and an optimal plan. For a uniform ellipsoidal target, disintegration over the submanifold gives conditional measures whose fiberwise $L<sup>\infty$-norms satisfy a sharp weighted average estimate with constant $ω<em>n/ω</em>{n+m}$ in codimensions $1\leq m\leq4$. This yields the sharp ellipsoidal Michael--Simon--Sobolev inequality in the same range, including its equality cases and symmetrization consequences. To the best of our knowledge, even in the Euclidean case, the sharp constants in codimensions three and four were previously unknown.