ACTA issues

Characterizing circles by a convex combinatorial property

Gábor Czédli

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

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.)