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

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## 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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