Bárány Imre: Same average in every direction

Given a polytope P in R^3 and a non-zero vector z, also in R^3, the plane with equation {x: zx=t}  (zx is the  calar product of z and x) intersects P in a convex polygon P(z,t) for all t in [t^-,t^+]. Here t^-=min {zx: x in P} and t^+=max {zx: x in P}. Let A(P,z) denote the average number of vertices of P(z,t) on the interval [t^-,t^+]. It is not hard to see that A(Q,z)=4 for every z in R^3 when Q is the unit cube. For what polytopes is A(P,z) a  constant independent of z? 
Joint work with Gabor Domokos.