Convex Bodies with Sheafs of Elliptic Sections II

A convex subset D of $E^d$ is an ellipsoid if all the sections given by hyperplanes are elliptic. We study whether we can restrict the hyperplanes to those that contain one of two fixed linear varieties. In a previous paper in this journal we considered the case where one of the varieties cuts the interior of D, and now we assume that the varieties support D.