Extension from ellipsoidal to general centrally symmetric targets

Extend the averaged fiberwise L∞ estimate and the associated ABP-based inequalities from centered ellipsoidal targets to general centrally symmetric convex targets.

Background

The transport construction itself applies to every bounded convex target and in every codimension, but the sharp fiber estimate proved in Theorem 1.2 relies on the ellipsoidal geometry of the target. Specifically, a linear change of variables converts each normal slice of a centered ellipsoid into a Euclidean ball, enabling the radial estimate used in the proof. The authors identify extending this step to general centrally symmetric targets as an unresolved direction.