ACTA issues

## Characterizing circles by a convex combinatorial property

Gábor Czédli

Acta Sci. Math. (Szeged) 83:3-4(2017), 683-701
70/2016

 Abstract. Let $K_0$ be a compact convex subset of the plane $\preal$, and assume that $K_1\subseteq\preal$ is similar to $K_0$, that is, $K_1$ is the image of $K_0$ with respect to a similarity transformation $\preal\to \preal$. Kira Adaricheva and Madina Bolat have recently proved that if $K_0$ is a disk and both $K_0$ and $K_1$ are contained in a triangle with vertices $A_0$, $A_1$, and $A_2$, then there exist a $j\in\set {0,1,2}$ and a $k\in\set {0,1}$ such that $K_{1-k}$ is contained in the convex hull of $K_k\cup(\set{A_0,A_1, A_2}\setminus\set {A_j})$. Here we prove that this property characterizes disks among compact convex subsets of the plane. In fact, 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. DOI: 10.14232/actasm-016-570-x AMS Subject Classification (1991): 52C99, 52A01 Keyword(s): convex hull, circle, abstract convex geometry, anti-exchange system, Carathéodory's theorem, carousel rule, boundary of a compact convex set, lattice Received December 13, 2016, and in revised form May 10, 2017. (Registered under 70/2016.)