A carousel property for compact convex sets

We prove that if $A_0$ and $A_1$ are compact convex 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 in the convex hull of $A_{1-i}$ and $\{G_1,\dots,G_n\}\setminus\{G_j\}$.

This result and its proof recover and generalize results of Adaricheva-Bolat and Czédli-Kurusa.

We also show that this bound is sharp for all $n$.

Our results yield explicit examples of convex geometries that cannot be represented by translates of convex bodies in the plane.