Imre Barany
	

Same average in every direction

Abstract: Given a polytope P in R^3 and a non-zero vector z  also in R^3, the plane {x\in R^3: zx=t}  (where zx is the scalar 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.