  Under what circumstances does the ``closeness" of two functions imply the
``closeness" of their respective sublevel sets? In this paper, we answer this
question by showing that if a sequence of functions converges strictly from
above/below to a function, $V$, in the $L^\infty$ (or $L^1$) norm then these
functions yield a sequence sublevel sets that converge to the sublevel set of
$V$ with respect to the Hausdorff metric (or volume metric). Based on these
theoretical results we propose Sum-of-Squares (SOS) numerical schemes for the
optimal outer/inner polynomial sublevel set approximation of various sets,
including intersections and unions of semialgebraic sets, Minkowski sums,
Pontryagin differences and discrete points. We present several numerical
examples demonstrating the usefulness of our proposed algorithm including
approximating sets of discrete points to solve machine learning one-class
classification problems and approximating Minkowski sums to construct C-spaces
for computing optimal collision-free paths for Dubin's car.
