Graphs and Combinatorics

, Volume 31, Issue 2, pp 361–392 | Cite as

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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Copyright information

© Springer Japan 2015

Authors and Affiliations

  • Greg Aloupis
    • 1
  • Luis Barba
    • 1
    • 2
  • 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

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