Same average in every direction

Abstract

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 $z$?