On the Total Perimeter of Homothetic Convex Bodies in a Convex Container

TL;DR

This paper investigates bounds on the total perimeter of homothetic convex bodies packed inside a convex container, establishing tight asymptotic bounds depending on geometric configurations and placement.

Contribution

It provides optimal bounds on the total perimeter of homothetic convex bodies in a convex container, considering various geometric arrangements and placements.

Findings

Total perimeter is O(log n) or O(1) depending on fit.

Bounds are tight up to constant factors.

Perimeter bounds depend on the parallelism of container and bodies.

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.