Why is Helfenstein's claim about equichordal points false?

TL;DR

This paper critically analyzes Helfenstein's 1956 claim about equichordal points, identifies a computational error, and explains why his method cannot resolve the problem, contrasting it with a later complete solution.

Contribution

It clarifies the invalidity of Helfenstein's approach and demonstrates that his method cannot be used to solve the equichordal point problem, despite some ongoing interest.

Findings

Identifies a computational error in Helfenstein's paper

Shows Helfenstein's method cannot solve the equichordal point problem

Highlights the correct solution by Rychlik (1997)

Abstract

This article explains why a paper by Heinz G. Helfenstein entitled "Ovals with equichordal points", published in J.London Math.Soc.31, 54-57, 1956, is incorrect. We point out a computational error which renders his conclusions invalid. More importantly, we explain that the method cannot be used to solve the equichordal point problem with the method presented there. Today, there is a solution to the problem: Marek R. Rychlik, "A complete solution to the equichordal point problem of Fujiwara, Blaschke, Rothe and Weizenb\"ock", Inventiones Mathematicae 129 (1), 141-212, 1997. However, some mathematicians still point to Helfenstein's paper as a plausible path to a simpler solution. We show that Helfenstein's method cannot be salvaged. The fact that Helfenstein's argument is not correct was known to Wirsing, but he did not explicitly point out the error. This article points out the error and the reasons for the failure of Helfenstein's approach in an accessible, and hopefully enjoyable way.