Same average in every direction

Imre Bárány, Gábor Domokos

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 \{z…