  We consider the problem of approximating a semialgebraic set with a
sublevel-set of a polynomial function. In this setting, it is standard to seek
a minimum volume outer approximation and/or maximum volume inner approximation.
As there is no known relationship between the coefficients of an arbitrary
polynomial and the volume of its sublevel sets, previous works have proposed
heuristics based on the determinant and trace objectives commonly used in
ellipsoidal fitting. For the case of star-convex semialgebraic sets, we propose
a novel objective which yields both an outer and an inner approximation while
minimizing the ratio of their respective volumes. This objective is
scale-invariant and easily interpreted. Numerical examples are given which show
that the approximations obtained are often tighter than those returned by
existing heuristics. We also provide methods for establishing the
star-convexity of a semialgebraic set by finding inner and outer approximations
of its kernel.
