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

TL;DR

This paper investigates the minimal convex polygon area containing a given number of congruent disks, providing bounds, constructions of optimal polygons, and exploring related geometric invariants and classical curves.

Contribution

It introduces a lower bound for the area of the smallest convex k-gon containing congruent disks and constructs optimal polygons in cases where the bound is tight.

Findings

Lower bound for polygon area using Wegner inequality

Construction of optimal polygons for specific cases

Analysis of geometric invariants and classical curve characterizations

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.