  We present proofs of the basic isopermetric structure theory, obtaining some
new simplified proofs. As an application, we obtain simple descriptions for
subsets $S$ of an abelian group with $|kS|\le k|S|-k+1$ or $|kS-rS|- (k+r)|S|,$
where $1\le r \le k.$ These results may be applied to several questions in
Combinatorics and Additive Combinatorics (Frobenius Problem, Waring's problem
in finite fields and Cayley graphs with a big diameter, ....).
