Old and new about equidistant sets and generalized conics

TL;DR

This paper reviews classical and new results on equidistant sets, explores their continuous dependence on focal sets, and proposes viewing them as generalized conics with geometric properties similar to classical conics.

Contribution

It introduces a new perspective on equidistant sets as generalizations of conics and proves their continuous variation with focal sets.

Findings

Equidistant sets vary continuously with their focal sets.

Many geometric features of classical conics are present in equidistant sets.

A shadowing property of equidistant sets is established with sharp estimates.

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.