Critical loci of convex domains in the plane (Q2020438)

In the tradition of Mahler, the authors study critical loci of planar domains. They point out that all previous examples of such sets were either finite sets or finite unions of closed curves. As an example, it is recalled that the critical locus of the open unit disc, \(D\), in the plane is homeomorphic to the unit circle. In Theorem 1.3 the authors show that there are bounded convex domains \(K\) in \(\mathbb{R}^2\) which are symmetric about the origin and whose critical locus is homeomorphic to a Cantor set. In particular, it is shown that each non-empty closed subset of the critical locus of the unit disc is the critical locus of some convex symmetric domain \(K \supset D\). Based on the concept of irreducibility, which was introduced by Mahler, the authors generalise this result. The main result of the paper is Theorem 1.8 which has Theorem 1.3 as a special case.