  Motivated by problems of uncertainty propagation and robust estimation we are
interested in computing a polynomial sublevel set of fixed degree and minimum
volume that contains a given semialgebraic set K. At this level of generality
this problem is not tractable, even though it becomes convex e.g. when
restricted to nonnegative homogeneous polynomials. Our contribution is to
describe and justify a tractable L^1-norm or trace heuristic for this problem,
relying upon hierarchies of linear matrix inequality (LMI) relaxations when K
is semialgebraic, and simplifying to linear programming (LP) when K is a
collection of samples, a discrete union of points.
