  The polynomial Fre\u{\i}man--Ruzsa conjecture is a fundamental open question
in additive combinatorics. However, over the integers (or more generally
$\mathbb{R}^d$ or $\mathbb{Z}^d$) the optimal formulation has not been fully
pinned down.
  The conjecture states that a set of small doubling is controlled by a very
structured set, with polynomial dependence of parameters. The ambiguity
concerns the class of structured sets needed. A natural formulation in terms of
generalized arithmetic progressions was recently disproved by Lovett and Regev.
A more permissive alternative is in terms of \emph{convex progressions}; this
avoids the obstruction, but uses is a significantly larger class of objects,
yielding a weaker statement.
  Here we give another formulation of PFR in terms of Euclidean ellipsiods (and
some variations). We show it is in fact equivalent to the convex progression
version; i.e. that the full range of convex progressions is not needed. The key
ingredient is a strong result from asymptotic convex geometry.
