  A classical theorem of Fritz John allows one to describe a convex body, up to
constants, as an ellipsoid. In this article we establish similar descriptions
for generalized (i.e. multidimensional) arithmetic progressions in terms of
proper (i.e. collision-free) generalized arithmetic progressions, in both
torsion-free and torsion settings. We also obtain a similar characterization of
iterated sumsets in arbitrary abelian groups in terms of progressions, thus
strengthening and extending recent results of Szemer\'edi and Vu.
