Sean Dewar

Homothetic packings of centrally symmetric convex bodies

A (2-dimensional) 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 disc. In this talk I will discuss the question of what types of contact graphs can arise from packings of homothetic copies of a single regular symmetric body C when some level of genericity is assumed for the packing's radii (i.e., the scalings of C). Oded Schramm proved in his thesis that any planar graph can be realised as the contact graph of a homothetic packing of a regular symmetric body, however the radii of such a packing is often non-generic. As will be shown in the talk, only (2,2)-sparse planar graphs (a type of graph with low sparsity) can arise from homothetic packings of regular symmetric convex bodies when the radii are chosen to be random. I will also discuss how the converse of this statement holds for almost all choices of centrally symmetric convex body C; i.e., for any planar (2,2)-sparse graph G, there is a positive measure set of radii which allow for packings with G as the contact graph. Time-depending, I will discuss recent work surrounding the special case of the square, a convex body that is neither smooth nor strictly convex.