Imre Bárány, Gábor Domokos: Same average in every direction, 125-138

Abstract:

Given a polytope $P\subset \mathbb{R}^3$ and a non-zero vector $z \in \mathbb{R}^3$, the plane $\{x\in \mathbb{R}^3:zx=t\}$ intersects $P$ in a 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 $z,x \in \mathbb{R}^3$. 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 $z$?

Key Words: Convex polytopes, zonotopes, number of vertices, fragmentation.

2020 Mathematics Subject Classification: Primary 52A15; Secondary 52A20.

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