# On convex closed planar curves as equidistant sets

## TL;DR

This paper proves that every convex closed planar curve can be represented as an equidistant set of two disjoint closed sets, generalizing the concept of convexity through equidistancy.

## Contribution

It demonstrates that the class of convex closed planar curves is exactly the set of equidistant sets of two disjoint closed sets, expanding the understanding of equidistancy as a convexity generalization.

## Key findings

- Any convex closed planar curve can be realized as an equidistant set.
- Equidistancy encompasses all convex closed planar curves.
- The result generalizes classical conics 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.