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 ${\rm per}(S)$, in terms of ${\rm per}(D)$ and $n$. When all homothets of $C$ touch the boundary of the container $D$, we show that either ${\rm per}(S)=O(\log n)$ or ${\rm per}(S)=O(1)$, depending on how $C$ and $D$ "fit together," and these bounds are the best possible apart from the constant factors. Specifically, we establish an optimal bound ${\rm per}(S)=O(\log n)$ unless $D$ is a convex polygon and every side of $D$ is parallel to a corresponding segment on the boundary of $C$ (for short, $D$ is \emph{parallel to} $C$). When $D$ is parallel to $C$ but the homothets of $C$ may lie anywhere in $D$, we show that ${\rm per}(S)=O((1+{\rm esc}(S)) \log n/\log \log n)$, where ${\rm esc}(S)$ 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.