Old and new about equidistant sets and generalized conics

Abstract

This article is devoted to the study of classical and new results concerning equidistant sets, both from the topological and metric point of view. We start with a review of the most interesting known facts about these sets in the euclidean space and then we prove that equidistant sets vary continuously with their focal sets. In the second part we propose a viewpoint in which equidistant sets can be thought of as natural generalization for conics. Along these lines, we show that many geometric features of classical conics can be retrieved in more general equidistant sets. In the Appendix we prove a shadowing property of equidistant sets and provide sharp estimates. This result should be of interest for computer simulations.