Carousel theorems for compact sets and homothets

Abstract

We prove that if $A_0$ and $A_1$ are compact sets contained in a convex $n$-gon with vertices $G_1, \dots, G_n$, and $2 \lceil \frac{n}{2} \rceil$ is strictly greater than the number of common supporting lines of $A_0$ and $A_1$, then there exist $i\in \{0,1\}$ and $j\in \{1, \dots, n\}$ such that $A_i$ is contained in the convex hull of $A_{1-i}$ and $\{G_1,\dots,G_n\}\setminus\{G_j\}$. This generalizes and recovers results of Adaricheva-Bolat and Czédli-Kurusa concerning disks and homothetic sets. We construct examples to prove that the bound is sharp. We also construct a family of convex geometries not representable by positive homothets of any fixed planar convex body.