Average pace and horizontal chords

TL;DR

This paper explores a mathematical problem related to running pace and horizontal chords, explaining why a certain intuitive assumption is false and providing historical context and streamlined proofs of related theorems.

Contribution

It clarifies the Universal Chord Theorem, explains its relevance to running pace problems, and streamlines a classical proof by Heinz Hopf from 1937.

Findings

The average pace does not guarantee a mile run in exactly that time.

The paper provides a clear explanation of the Universal Chord Theorem.

It offers a streamlined proof of Heinz Hopf's result from 1937.

Abstract

We are motivated by a problem about running: If a race was completed in an average pace of P minutes per mile, is there necessarily some mile of the race that was run in exactly P minutes? The answer is no. We explain why, and describe the history of this celebrated problem, known as the Universal Chord Theorem. We also clarify and streamline the proof of a more powerful result by Heinz Hopf from 1937.