Homothetic packings of centrally symmetric convex bodies

TL;DR

This paper investigates the geometric and combinatorial properties of homothetic packings of centrally symmetric convex bodies, establishing conditions for their contact graphs and their realizability.

Contribution

It proves that homothetic packings of regular symmetric bodies have planar contact graphs with specific sparsity, and characterizes the set of bodies for which all such graphs can be realized.

Findings

Homothetic packings have (2,2)-sparse planar contact graphs.

Existence of a large set of bodies where all (2,2)-sparse graphs are realizable.

Planar contact graphs of these packings exhibit specific sparsity properties.

Abstract

A centrally symmetric convex body is a convex compact set with non-empty interior that is symmetric about the origin. Of particular interest are those that are both smooth and strictly convex -- known here as regular symmetric bodies -- since they retain many of the useful properties of the $d$-dimensional Euclidean ball. We prove that for any given regular symmetric body $C$, a homothetic packing of copies of $C$ with randomly chosen radii will have a $(2,2)$-sparse planar contact graph. We further prove that there exists a comeagre set of centrally symmetric convex bodies $C$ where any $(2,2)$-sparse planar graph can be realised as the contact graph of a stress-free homothetic packing of $C$.