  The well--known Freiman--Ruzsa Theorem provides a structural description of a
set $A$ of integers with $|2A|\le c|A|$ as a subset of a $d$--dimensional
arithmetic progression $P$ with $|P|\le c'|A|$, where $d$ and $c'$ depend only
on $c$. The estimation of the constants $d$ and $c'$ involved in the statement
has been the object of intense research. Freiman conjectured in 2008 a formula
for the largest volume of such a set. In this paper we prove the conjecture for
a general class of sets called chains.
