2012/01/11 by Mario Ponce, Ponce, Mario, Patricio Santibáñez +1
Mathematics · Physics and Astronomy · #51-02 #Advanced Differential Geometry Research #FOS: Mathematics #Mathematics and Applications #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1201.2582
openalex publication_date 2012/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.