  We show that, for a positive integer $r$, every minimal 1-saturating set in
${\rm PG}(r-1,2)$ of size at least ${11/36} 2^r+3$ is either a complete cap or
can be obtained from a complete cap $S$ by fixing some $s\in S$ and replacing
every point $s'\in S\setminus\{s\}$ by the third point on the line through $s$
and $s'$. Stated algebraically: if $G$ is an elementary abelian 2-group and a
set $A\subseteq G\setminus\{0\}$ with $|A|>{11/36} |G|+3$ satisfies $A\cup
2A=G$ and is minimal subject to this condition, then either $A$ is a maximal
sum-free set, or there are a maximal sum-free set $S\subseteq G$ and an element
$s\in S$ such that $A=\{s\}\cup\big(s+(S\setminus\{s\})\big)$. Since,
conversely, every set obtained in this way is a minimal 1-saturating set, and
the structure of large sum-free sets in an elementary 2-group is known, this
provides a complete description of large minimal 1-saturating sets.
  Our approach is based on characterizing those large sets $A$ in elementary
abelian 2-groups such that, for every proper subset $B$ of $A$, the sumset 2B
is a proper subset of 2A.
