  Recently proposed as a stable means of evaluating geometric compactness, the
isoperimetric profile of a planar domain measures the minimum perimeter needed
to inscribe a shape with prescribed area varying from 0 to the area of the
domain. While this profile has proven valuable for evaluating properties of
geographic partitions, existing algorithms for its computation rely on
aggressive approximations and are still computationally expensive. In this
paper, we propose a practical means of approximating the isoperimetric profile
and show that for domains satisfying a "thick neck" condition, our
approximation is exact. For more general domains, we show that our bound is
still exact within a conservative regime and is otherwise an upper bound. Our
method is based on a traversal of the medial axis which produces efficient and
robust results. We compare our technique with the state-of-the-art
approximation to the isoperimetric profile on a variety of domains and show
significantly tighter bounds than were previously achievable.
