The problem of reconstructing a geometric object from its moments is of interest in a variety of fields. In the paper The Inverse Moment Problem for Convex Polytopes, the authors show that a convex polytope $P$ with $m$ vertices can be uniquely determined by its moments up to order $2m - 1$.
My question is: how can we, knowing say $(2m - 1) - n$ moments, quantify how "far away" members of the resultant space of potential reconstructions are from the true one?
Any suggestions or pointers to existing results would be much appreciated!
