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

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.