  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.
