Why is Helfenstein's claim about equichordal points false?

Marek Rychlik

Published 2012-02-08Version 1

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.