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

• 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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