On the total perimeter of homothetic convex bodies in a convex container

  • Adrian Dumitrescu
  • Csaba D. Tóth
Original Paper


For two planar convex bodies, \(C\) and \(D\), consider a packing \(S\) of \(n\) positive homothets of \(C\) contained in \(D\). We estimate the total perimeter of the bodies in \(S\), denoted \(\mathrm{per}(S)\), in terms of \(\mathrm{per}(D)\) and \(n\). When all homothets of \(C\) touch the boundary of the container \(D\), we show that either \(\mathrm{per}(S)=O(\log n)\) or \(\mathrm{per}(S)=O(1)\), depending on how \(C\) and \(D\) “fit together”. Apart from the constant factors, these bounds are the best possible. Specifically, we prove that \(\mathrm{per}(S)=O(1)\) if \(D\) is a convex polygon and every side of \(D\) is parallel to a corresponding segment on the boundary of \(C\) (for short, \(D\) is parallel to \(C\)) and \(\mathrm{per}(S)=O(\log n)\) otherwise. When \(D\) is parallel to \(C\) but the homothets of \(C\) may lie anywhere in \(D\), we show that \(\mathrm{per}(S)=O((1+\mathrm{esc}(S)) \log n/\log \log n)\), where \(\mathrm{esc}(S)\) denotes the total distance of the bodies in \(S\) from the boundary of \(D\). Apart from the constant factor, this bound is also the best possible.


Convex body Perimeter Maximum independent set  Homothet  Ford disks Traveling salesman Approximation algorithm 

Mathematics Subject Classification

05B40 52C15 52C26 



We are grateful to Guangwu Xu for kindly showing us a short alternative derivation of the number theoretical sum in Eq. (5). We also thank an anonymous reviewer for his very careful reading of the manuscript and his pertinent remarks.


  1. Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1976)zbMATHGoogle Scholar
  2. Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM 45(5), 753–782 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bezdek, K.: On a strong version of the Kepler conjecture. Mathematika 59(1), 23–30 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bern, M., Eppstein, D.: Approximation algorithms for geometric problems. In: Hochbaum, D.S. (ed.) Approximation Algorithms for NP-hard Problems, pp. 296–345. PWS Publishing Company, Boston (1997)Google Scholar
  5. Brass, P., Moser, W.O.J., Pach, J.: Research Problems in Discrete Geometry. Springer, New York (2005)zbMATHGoogle Scholar
  6. de Berg, M., Gudmundsson, J., Katz, M.J., Levcopoulos, C., Overmars, M.H., van der Stappen, A.F.: TSP with neighborhoods of varying size. J. Algorithms 57(1), 22–36 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Dumitrescu, A., Mitchell, J.S.B.: Approximation algorithms for TSP with neighborhoods in the plane. J. Algorithms 48(1), 135–159 (2003)Google Scholar
  8. Dumitrescu, A., Tóth, C.D.: Minimum weight convex Steiner partitions. Algorithmica 60(3), 627–652 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Dumitrescu, A., Tóth, C.D.: The traveling salesman problem for lines, balls and planes. In: Proceedings of 24th ACM-SIAM Symposium on Discrete Algorithms, 2013, SIAM, pp. 828–843Google Scholar
  10. Ford, K.R.: Fractions. Am. Math. Mon. 45(9), 586–601 (1938)CrossRefGoogle Scholar
  11. Glazyrin, A., Morić, F.: Upper bounds for the perimeter of plane convex bodies. Acta Mathematica Hungarica 142(2), 366–383 (2014)Google Scholar
  12. Graham, R.L., Lagarias, J.C., Mallows, C.L., Wilks, A.R., Yan, C.H.: Apollonian circle packings: geometry and group theory I: the Apollonian group. Discret. Comput. Geom. 34, 547–585 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Hales, T.C.: The strong dodecahedral conjecture and Fejes Tóth’s conjecture on sphere packings with kissing number twelve. In: Discrete Geometry and Optimization, vol. 69 of Fields Communications, pp. 121–132, Springer, Switzerland (2013)Google Scholar
  14. Levcopoulos, C., Lingas, A.: Bounds on the length of convex partitions of polygons. In: Proceedings of 4th Annual Conference on Foundations of Software Technology and Theoretical Computer Science, LNCS 181, pp. 279–295, Springer (1984)Google Scholar
  15. Mata, C., Mitchell, J.S.B.: Approximation algorithms for geometric tour and network design problems. In: Proceedings of 11th ACM Symposium on Computational Geometry, ACM, pp. 360–369 (1995)Google Scholar
  16. Mitchell, J.S.B.: A constant-factor approximation algorithm for TSP with pairwise-disjoint connected neighborhoods in the plane. In: Proceedings of 26th ACM Symposium on Computational Geometry, ACM, pp. 183–191 (2010)Google Scholar

Copyright information

© The Managing Editors 2014

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of Wisconsin–MilwaukeeMilwaukeeUSA
  2. 2.Department of MathematicsCalifornia State University, NorthridgeLos AngelesUSA

Personalised recommendations