Cutting convex curves

TL;DR

The paper proves that for any two convex curves with opposite orientations in Euclidean space, there exists a hyperplane separating their corresponding points at each parameter, based on a general equipartition result.

Contribution

It introduces a novel geometric separation theorem for convex curves with opposite orientations using hyperplanes and equipartitions.

Findings

Existence of a separating hyperplane for convex curves with opposite orientations.

Generalization to equipartitions of ordered point sets.

Application to convex geometry and hyperplane arrangements.

Abstract

We show that for any two convex curves $C_1$ and $C_2$ in $\mathbb R^d$ parametrized by $[0,1]$ with opposite orientations, there exists a hyperplane $H$ with the following property: For any $t\in [0,1]$ the points $C_1(t)$ and $C_2(t)$ are never in the same open halfspace bounded by $H$. This will be deduced from a more general result on equipartitions of ordered point sets by hyperplanes.