The smallest convex K-gon containing N congruent disks

2021/02/04 by Erdenebaatar, Orgil-Erdene, Ninjbat, Uuganbaatar
#52A27 #52C15 #90C27 #Combinatorics (math.CO) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2102.02568

Abstract

Consider the problem of fnding the smallest area convex k-gon containing n∈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.

Related

• On the smallest area n-1-gon containing a convex n-gon
• An algorithm to find maximum area polygons circumscribed about a convex\n polygon
• Covering Convex Polygons by Two Congruent Disks
• Dowker-type theorems for disk-polygons in normed planes
• The intersection graph of the disks with diameters the sides of a convex n-gon