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Abstract
We study the set of lengths of the horizontal chords of a continuous function. We give a new proof of Hopf’s characterization of this set, and show that it implies that no matter which function we choose, at least half of the possible lengths occur. We prove several results about functions for which all the possible lengths occur.
MSC:
Acknowledgment
We thank CIRM for an excellent working environment during our « recherche en binomes » collaboration in July 2022. We used Desmos [Citation6] to experiment, explore all of the many cases, and create our figures. We thank the referees, whose suggestions improved our article.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Additional information
Notes on contributors
Diana Davis
Diana Davis received her B.A. from Williams College in 2007 and her Ph.D. from Brown University in 2013. She enjoys running, traveling, sailing, and holding hands with her wife.
Phillips Exeter Academy
Postal address: 20 Main Street, Exeter, NH 03833 USA
Serge Troubetzkoy
Serge E. Troubetzkoy received his B.A. from Yale University in 1982 and his Ph.D. from Stanford University in 1987. After a Brownian career path: Leningrad, Toronto, Warwick, Bielefeld, Stony Brook, and UAB, he settled down in Marseille.
Aix Marseille Univ, CNRS, I2M, Marseille, France
Postal address: I2M, Luminy, Case 907, F-13288 Marseille Cedex 9, France