On convex closed planar curves as equidistant sets

Abstract

The equidistant set of two nonempty subsets $K$ and $L$ in the Euclidean plane is a set all of whose points have the same distance from $K$ and $L$. Since the classical conics can be also given in this way, equidistant sets can be considered as a kind of their generalizations: $K$ and $L$ are called the focal sets. In their paper \cite{PS} the authors posed the problem of the characterization of closed subsets in the Euclidean plane that can be realized as the equidistant set of two connected disjoint closed sets. We prove that any convex closed planar curve can be given as an equidistant set, i.e. the set of equidistant curves contains the entire class of convex closed planar curves. In this sense the equidistancy is a generalization of the convexity.