Graphs and Combinatorics

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

Isoperimetric Enclosures

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


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.


Isoperimetric problems Voronoi diagram Enclosing curves Optimization problems 


  1. 1.
    Aggarwal, A., Guibas, L., Saxe, J., Shor, P.: A linear time algorithm for computing the Voronoi diagram of a convex polygon. In: Proceedings of STOC, pp. 39–45. ACM, New York (1987)Google Scholar
  2. 2.
    Aggarwal, A., Guibas, L.J., Saxe, J., Shor, P.W.: A linear-time algorithm for computing the voronoi diagram of a convex polygon. Discret. Comput. Geom. 4(1), 591–604 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Baker, A.: Transcendental Number Theory. Cambridge University Press, London (1990)Google Scholar
  4. 4.
    Banchoff, T., Giblin, P.: On the geometry of piecewise circular curves. Am. Math. Mon. 101(5), 403–416 (1994)Google Scholar
  5. 5.
    Barba, L.: Disk constrained 1-center queries. In: Proceedings of the Canadian Conference on Computational Geometry, CCCG, vol. 12, pp. 15–19 (2012)Google Scholar
  6. 6.
    Besicovitch, A.S.: Variants of a classical isoperimetric problem. Q. J. Math. Oxf. Ser. (2) 3, 42–49 (1952)Google Scholar
  7. 7.
    Blåsjö, V.: The isoperimetric problem. Am. Math. Mon. 112(6), 526–566 (2005)CrossRefzbMATHGoogle Scholar
  8. 8.
    Borwein, J.M., Crandall, R.E.: Closed forms: what they are and why we care. Not. Am. Math. Soc. 60(1), 50–65 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Bose, P., De Carufel, J.-L.: Isoperimetric triangular enclosures with a fixed angle. J. Geom. 104(2), 229–255 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Chandrasekaran, K., Dadush, D., Vempala, S.: Thin partitions: isoperimetric inequalities and a sampling algorithm for star shaped bodies. In: SODA, pp. 1630–1645 (2010)Google Scholar
  11. 11.
    Chow, T.Y.: What is a closed-form number? Am. Math. Mon. 106(5), 440–448 (1999)CrossRefzbMATHGoogle Scholar
  12. 12.
    de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Berlin (2008)Google Scholar
  13. 13.
    Garey, M.R., Graham, R.L., Johnson, D.S.: The complexity of computing Steiner minimal trees. SIAM J. Appl. Math. 32(4), 835–859 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Hwang, F.K., Richards, D.S., Winter, P.: The Steiner Tree Problem. Elsevier, Amsterdam (1992)Google Scholar
  15. 15.
    Koutsoupias, E., Papadimitriou, C.H., Sideri, M.: On the optimal bisection of a polygon. INFORMS J. Comput. 4(4), 435–438 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Megiddo, N.: Linear-time algorithms for linear programming in \(\mathbb{R}^3\) and related problems. SIAM J. Comput. 12(4), 759–776 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Morgan, F., Hutchings, M., Howards, H.: The isoperimetric problem on surfaces of revolution of decreasing gauss curvature. Trans. Am. Math. Soc. 352(11), 4889–4909 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Polya, G.: Mathematics and Plausible Reasoning: Induction and Analogy in Mathematics, vol. 1. Princeton University Press, New Jersey (1990)Google Scholar
  19. 19.
    Preparata, F., Shamos, M.: Computational Geometry—An Introduction. Springer, Berlin (1985)Google Scholar
  20. 20.
    Preparata, F.P.: The medial axis of a simple polygon. In: Mathematical Foundations of Computer Science 1977, pp. 443–450. Springer, Berlin (1977)Google Scholar
  21. 21.
    Singer, D.A.: Geometry: Plane and Fancy. Springer, Berlin (1998)Google Scholar
  22. 22.
    Welzl, E.: Smallest enclosing disks (balls and ellipsoids). In: Results and New Trends in Computer Science, pp. 359–370. Springer, Berlin (1991)Google Scholar

Copyright information

© Springer Japan 2015

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

Personalised recommendations