A Carousel Property for Compact Convex Sets

TL;DR

This paper establishes a new geometric property relating compact convex sets within polygons, generalizing previous results and demonstrating the sharpness of bounds for even-sided polygons.

Contribution

It introduces a novel carousel property for convex sets in polygons, extending prior theorems and providing sharp bounds for even-sided polygons.

Findings

Proves a new property for convex sets in polygons with more than the number of common supporting lines.

Generalizes previous results by Adaricheva--Bolat and Czédli.

Shows the bound is sharp for polygons with an even number of sides.

Abstract

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 $n$ 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 recovers and generalizes previous results of Adaricheva--Bolat and Cz{\'e}dli. We also show that this bound is sharp for even $n$.