Critical loci of convex domains in the plane

Let K be a bounded convex domain in R2 symmetric about the origin. The critical locus of is defined to (non-empty compact) set lattices Λ smallest possible covolume such that Λ∩K={0}. These are classical objects geometry numbers; yet all previously known examples loci were either finite sets or unions closed curves. In this paper we give new construction which, particular, furnishes domains having arbitrary Hausdorff dimension between 0 and 1.