Thick sections of convex bodies

Let P and Q   be origin-symmetric convex bodies in RnRn such that for every slab (thick section) of a fixed width 2t, symmetric about the origin, its intersections with P and Q   have equal volumes. Is then necessarily P=QP=Q? We show that this is true in the class of origin-symmetric convex polytopes. We also study a modified version of this problem, when one of the bodies is a Euclidean ball and both bodies have the same volume. Finally we discuss a related problem concerning sections of convex bodies by hyperplanes that are distance t from the origin.