Title: On the generalization of Erdős-Vincze's theorem about the approximation of closed convex plane curves by 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 Erdős-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.
Vázsonyi (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 Erdős-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.