On the generalization of Erd\H{o}s-Vincze's theorem about the approximation of closed convex plane curves by polyellipses

TL;DR

This paper extends Erd ext{"o}s-Vincze's theorem, showing that regular polygons can be approximated by polyellipses as the number of vertices increases, and explores the limits of polyellipses in approximating convex curves.

Contribution

It generalizes Erd ext{"o}s-Vincze's theorem from triangles to regular polygons and investigates the approximation of convex curves by polyellipses using Hausdorff limits.

Findings

Approximation error tends to zero as polygon vertices increase.

Regular polygons can be approximated by polyellipses in the Hausdorff sense.

Curves with constant average distance from focal sets can be limits of polyellipses.

Abstract

A polyellipse is a curve in the Euclidean plane all of whose points have the same sum of distances from finitely many given points (focuses). The classical version of Erd\H{o}s-Vincze's theorem states that regular triangles can not be presented as the Hausdorff limit of polyellipses even if the number of the focuses can be arbitrary large. In other words the topological closure of the set of polyellipses with respect to the Hausdorff distance does not contain any regular triangle and we have a negative answer to the problem posed by E. V\'azsonyi (Weissfeld) about the approximation of closed convex plane curves by polyellipses. It is the additive version of the approximation of simple closed plane curves by polynomial lemniscates all of whose points have the same product of distances from finitely many given points (focuses). Here we are going to generalize the classical version of Erd\H{o}s-Vincze's theorem for regular polygons in the plane. We will conclude that the error of the approximation tends to zero as the number of the vertices of the regular polygon tends to the infinity. The decreasing tendency of the approximation error gives the idea to construct curves in the topological closure of the set of polyellipses. If we use integration to compute the average distance of a point from a given (focal) set in the plane then the curves all of whose points have the same average distance from the focal set can be given as the Hausdorff limit of polyellipses corresponding to partial sums.