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

  • Adrian Dumitrescu
  • Csaba D. Tóth
Original Paper

Abstract

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.

Keywords

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

Mathematics Subject Classification

05B40 52C15 52C26 

Notes

Acknowledgments

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.

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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

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