Medial Axis Isoperimetric Profiles

TL;DR

This paper introduces an efficient method for approximating the isoperimetric profile of planar domains, providing exact results for certain shapes and tighter bounds than existing methods, with practical applications in geometric analysis.

Contribution

The authors propose a medial axis-based approximation technique for the isoperimetric profile that is both practical and more accurate than previous algorithms, especially for domains with a 'thick neck' condition.

Findings

Exact approximation for domains with a 'thick neck'

Tighter bounds compared to state-of-the-art methods

Efficient and robust medial axis traversal approach

Abstract

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.