Inextensible Domains

We develop a theory of planar, origin-symmetric, convex domains that are inextensible with respect to lattice covering, that is, domains such that augmenting them in any way allows fewer domains to cover the same area. We show that originsymmetric inextensible domains are exactly the origin-symmetric convex domains with a circle of outer billiard triangles. We address a conjecture by Genin and Tabachnikov about convex domains, not necessarily symmetric, with a circle of outer billiard triangles, and show that it follows immediately from a result of Sas. In a series of papers from 1946 to 1947, Kurt Mahler developed a theory of planar, origin-symmetric, star-like domains that are irreducible with respect to lattice packing, that is, domains such that taking away any piece allows more domains to be packed in the same area [5, 6]. In particular, he was interested in the problem of identifying convex domains such that every convex domain with lower area may be packed at a greater number per unit area than the domain in question. This is the subject of Reinhardt’s conjecture [7]. In particular, Mahler showed that the disk is not such a domain, even if the domains of lower area to which we compare it are restricted to a small neighborhood of the disk with respect to Hausdorff distance [6]. Inspired by the work of Mahler, we develop a theory of planar, originsymmetric, convex domains that are inextensible with respect to lattice covering, that is, domains such that adding any piece allows fewer domains to cover the same area. We find that the inextensible domains are simply those with a circle of outer billiard triangles, a family of domains studied previously by Genin and Tabachnikov [3]. The analogue of Reinhardt’s domain for covering is simply the ellipse: any originsymmetric convex domain can cover the plane with no larger number per unit area than an ellipse of the same area, as can be easily shown using a classical result of Sas [8, 9]. In other words, the ellipse covers the plane with the least efficiency. Genin and Tabachnikov conjecture that out of convex domains, not necessarily symmetric, with a circle of critical triangles of a fixed area, Date: November 30, 2013.