Set approximation via minimum-volume polynomial sublevel sets

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 L¹-norm or trace heuristic for this problem, relying upon hierarchies of linear matrix inequality (LMI) relaxations when K is semialgebraic, and simplifying to linear constraints when K is a collection of samples, a discrete union of points.