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Under what circumstances does the ``closeness" of two functions imply the``closeness" of their respective sublevel sets? In this paper, we answer thisquestion by showing that if a sequence of functions converges strictly fromabove/below to a function, $V$, in the $L^\infty$ (or $L^1$) norm then thesefunctions yield a sequence sublevel sets that converge to the sublevel set of$V$ with respect to the Hausdorff metric (or volume metric). Based on thesetheoretical results we propose Sum-of-Squares (SOS) numerical schemes for theoptimal 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 numericalexamples demonstrating the usefulness of our proposed algorithm includingapproximating sets of discrete points to solve machine learning one-classclassification problems and approximating Minkowski sums to construct C-spacesfor computing optimal collision-free paths for Dubin's car.

Sublevel Set Approximation in The Hausdorff and Volume Metric with Application to Path Planning and Obstacle Avoidance