Critical loci of convex domains in the plane

Abstract

Let $K$ be a bounded convex domain in ${\mathbb{R}}^2$ symmetric about the origin. The critical locus of $K$ is defined to be the (non-empty compact) set of lattices $\Lambda$ in ${\mathbb{R}}^2$ of smallest possible covolume such that $\Lambda \cap K=\{0\}$. These are classical objects in geometry of numbers; yet all previously known examples of critical loci were either finite sets or finite unions of closed curves. In this paper we give a new construction which, in particular, furnishes examples of domains having critical locus of arbitrary Hausdorff dimension between $0$ and $1$.

new section added