The smallest convex $K$-gon containing $N$ congruent disks

Orgil-Erdene Erdenebaatar; Uuganbaatar Ninjbat

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arXiv · 2102.02568

The smallest convex $K$-gon containing $N$ congruent disks

Orgil-Erdene Erdenebaatar·Uuganbaatar Ninjbat

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Abstract

Consider the problem of fnding the smallest area convex $k$-gon containing $n\in\mathbb{N}$ congruent disks without an overlap. By using Wegner inequality in sphere packing theory we give a lower bound for the area of such polygons. For several cases where this bound is tight we construct corresponding optimal polygons. We also discuss its solution for some cases where this bound is not tight, e.g. $n = 2$ and $k$ is odd, and $n = 3$; $k = 4$. On the way to prove our results we prove a result on geometric invariants between two polygons whose sides are pairwise parallel, and give a new characterisation for the trisectrix of Maclaurin.

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Orgil-Erdene Erdenebaatar, Uuganbaatar Ninjbat. 2021-02-04. The smallest convex $K$-gon containing $N$ congruent disks. https://arxiv.org/abs/2102.02568

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