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On the generalization of Erd\H{o}s-Vincze's theorem about the approximation of closed convex plane curves by polyellipses by Vincze, Csaba; Kovács, Zoltán; Csorvássy, Zsófia Fruzsina is a scholarly article available to read on EtoBox.

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

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 topolog

Author

Vincze, Csaba; Kovács, Zoltán; Csorvássy, Zsófia Fruzsina

Published

2017

Language

EN