Same average in every direction

TL;DR

This paper investigates the conditions under which the average number of vertices of plane intersections with a polytope remains constant regardless of the direction of the intersecting plane.

Contribution

It characterizes polytopes for which the average number of vertices in plane intersections is direction-independent.

Findings

Identifies polytopes with constant average vertices across all directions.

Provides criteria for polytopes with uniform intersection properties.

Explores geometric conditions for directional invariance.

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 $z$?