On the Total Perimeter of Homothetic Convex Bodies in a Convex Container
Adrian Dumitrescu, Csaba D. T\'oth

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, and , consider a packing of positive homothets of contained in . We estimate the total perimeter of the bodies in , denoted , in terms of and . When all homothets of touch the boundary of the container , we show that either or , depending on how and "fit together," and these bounds are the best possible apart from the constant factors. Specifically, we establish an optimal bound unless is a convex polygon and every side of is parallel to a corresponding segment on the boundary of (for short, is \emph{parallel to} ). When is parallel to but the homothets of may lie anywhere in , we show that , where denotes the total…
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