  Given a subset $W$ of an abelian group $G$, a subset $C$ is called an
additive complement for $W$ if $W+C=G$; if, moreover, no proper subset of $C$
has this property, then we say that $C$ is a minimal complement for $W$. It is
natural to ask which subsets $C$ can arise as minimal complements for some $W$.
We show that in a finite abelian group $G$, every non-empty subset $C$ of size
$|C| \leq 2^{2/3}|G|^{1/3}/((3e \log |G|)^{2/3}$ is a minimal complement for
some $W$. As a corollary, we deduce that every finite non-empty subset of an
infinite abelian group is a minimal complement. We also derive several
analogous results for ``dual'' problems about maximal supplements.
