 

Marek Rychlik
Why is Helfenstein's claim about equichordal points false? view print


Published: 
June 28, 2012 
Keywords: 
Equichordal point problem, convex geometry 
Subject: 
52A10, 34K19 


Abstract
This article explains why a paper by Heinz G. Helfenstein entitled
Ovals with equichordal points, J. London
Math. Soc. 31 (1956), 5457, is incorrect. We point out a
computational error which renders his conclusions invalid. More
importantly, we explain that the method presented there cannot be used to solve the
equichordal point problem. Today,
there is a solution to the problem: Marek R. Rychlik, A complete
solution to the equichordal point problem of Fujiwara, Blaschke,
Rothe and Weizenböck, Inventiones Mathematicae 129 (1997),
141212. 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.


Author information
University of Arizona, Department of Mathematics, 617 N Santa Rita Rd, P.O. Box 210089, Tucson, AZ 857210089, USA

