Characterizing circles by a convex combinatorial property

TL;DR

This paper characterizes circles among convex sets in the plane by a property involving inclusion relations of similar or congruent sets within triangles, extending previous results from disks to all convex sets.

Contribution

It proves that a specific convex combinatorial property characterizes circles among convex sets, generalizing prior results from disks to all convex sets using isometric transformations.

Findings

The property characterizes circles among convex sets.

The result extends previous characterizations from disks to all convex sets.

Circles are uniquely identified by this convex combinatorial property.

Abstract

Let $K_0$ be a compact convex subset of the plane $\mathbb R^2$, and assume that $K_1\subseteq \mathbb R^2$ is similar to $K_0$, that is, $K_1$ is the image of $K_0$ with respect to a similarity transformation $\mathbb R^2\to\mathbb R^2$. Kira Adaricheva and Madina Bolat have recently proved that if $K_0$ is a disk and both $K_0$ and $K_1$ are included in a triangle with vertices $A_0$, $A_1$, and $A_2$, then there exist a $j\in \{0,1,2\}$ and a $k\in\{0,1\}$ such that $K_{1-k}$ is included in the convex hull of $K_k\cup(\{A_0,A_1, A_2\}\setminus\{A_j\})$. Here we prove that this property characterizes disks among compact convex subsets of the plane. Actually, we prove even more since we replace "similar" by "isometric" (also called "congruent"). Circles are the boundaries of disks, so our result also gives a characterization of circles.