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Abstract Recently proposed as a stable means of evaluating geometric compactness, theisoperimetric profileof 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.
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Isoperimetry in Surfaces of Revolution with Density
The isoperimetric problem with a density or weighting seeks to enclose prescribed weighted volume with minimum weighted perimeter. According to Chambers' recent proof of the log-convex density conjecture, for many densities on Rn the answer is a sphere about the origin. We seek to generalize his results to some other spaces of revolution or to two di erent densities for volume and perimeter. We provide general results on existence and boundedness and a new approach to proving circles about the origin isoperimetric.
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- Award ID(s):
- 1659037
- PAR ID:
- 10092515
- Date Published:
- Journal Name:
- Missouri journal of mathematical sciences
- Volume:
- 30
- Issue:
- 2
- ISSN:
- 0899-6180
- Page Range / eLocation ID:
- 150-165
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- The DOI is not currently available.
