Graphs and Combinatorics

, Volume 31, Issue 2, pp 361–392

Isoperimetric Enclosures

• Greg Aloupis
• Luis Barba
• Jean-Lou De Carufel
• Stefan Langerman
• Diane L. Souvaine
Original Paper

Abstract

Given a number $$P$$, we study the following three isoperimetric problems introduced by Besicovitch in 1952: (1) Let $$S$$ be a set of $$n$$ points in the plane. Among all the curves with perimeter $$P$$ that enclose $$S$$, what is the curve that encloses the maximum area? (2) Let $$Q$$ be a convex polygon with $$n$$ vertices. Among all the curves with perimeter $$P$$ contained in $$Q$$, what is the curve that encloses the maximum area? (3) Let $$r_{\circ }$$ be a positive number. Among all the curves with perimeter $$P$$ and circumradius $$r_{\circ }$$, what is the curve that encloses the maximum area? In this paper, we provide a complete characterization for the solutions to Problems 1, 2 and 3. We show that there are cases where the solution to Problem 1 cannot be computed exactly. However, it is possible to compute in $$O(n \log n)$$ time the exact combinatorial structure of the solution. In addition, we show how to compute an approximation of this solution with arbitrary precision. For Problem 2, we provide an $$O(n\log n)$$-time algorithm to compute its solution exactly. In the case of Problem 3, we show that the problem can be solved in constant time. As a side note, we show that if $$S$$ is a set of $$n$$ points in the plane, then finding the area of the curve of perimeter $$P$$ that encloses $$S$$ and has minimum area is NP-hard.

Keywords

Isoperimetric problems Voronoi diagram Enclosing curves Optimization problems

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Authors and Affiliations

• Greg Aloupis
• 1
• Luis Barba
• 1
• 2
Email author
• Jean-Lou De Carufel
• 2
• Stefan Langerman
• 1
• Diane L. Souvaine
• 3
1. 1.Département d’InformatiqueUniversité Libre de BruxellesBrusselsBelgium
2. 2.School of Computer ScienceCarleton UniversityOttawaCanada
3. 3.Department of Computer ScienceTufts UniversityMedfordUSA