  Given a polytope $P\subset R^3$ and a non-zero vector $z \in R^3$, the plane
$\{x\in R^3:zx=t\}$ intersects $P$ in convex polygon $P(z,t)$ for $t \in
[t^-,t^+]$ where $t^-=\min \{zx: x \in P\}$ and $t^+=\max \{zx: x\in P\}$, $zx$
is the scalar product of $z,x \in R^3$. Let $A(P,z)$ denote the average number
of vertices of $P(z,t)$ on the interval $[t^-,t^+]$. For what polytopes is
$A(P,z)$ a constant independent of $z$?
