Midconvex sets in Abelian groups

Abstract

A subset $X$ of an Abelian group $G$ is called $midconvex$ if for every $x,y\in X$ the set $\frac{x+y}{2}=\{z\in G:2z=x+y\}$ is a subset of $X$. We prove that a subset $X$ of an Abelian group $G$ is midconvex if and only if for every $g\in G$ and $x\in X$, the set $\{n\in \mathbb{Z}:x+ng\in X\}$ is equal to $C\cap H$ for some order-convex set $C\subseteq \mathbb{Z}$ and some subgroup $H\subseteq \mathbb{Z}$ such that the quotient group $\mathbb{Z}/H$ has no elements of even order. This characterization implies that a subset $X$ of a periodic Abelian group $G$ is midconvex if and only if for every $x\in X$ the set $X-x$ is a subgroup of $G$ such that every element of the quotient group $G/(X-x)$ has odd order. Also we prove that a nonempty set $X$ in a subgroup $G\subseteq \mathbb{Q}$ is midconvex if and only if $X=C\cap (H+x)$ for some order-convex set $C\subseteq \mathbb{Q}$, some $x\in X$ and some subgroup $H$ of $G$ such that the quotient group $G/H$ contains no elements of even order.

4 pages