Meetup summary
2026-03-06 - Universal chord theorem
Recommended reading:
- Average pace and horizontal chords by Keith Burns, Orit Davidovich and Diana Davis
Agenda:
Neil will be covering the universal chord theorem, which can be seen as a generalization of Rolle’s theorem in some sense. This will be a self-contained session. The problem came up while Neil and I were running a 5k. We were arguing as to whether you necessarily ran at least a contiguous mile at or above your average 5k pace by the time of completion.1 The universal chord theorem and paper answer this question definitively.
Footnotes
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Spoiler warning: We made some hand-wavy arguments as to why you would probably have run a contiguous mile at pace, invoking the mean value theorem/intermediate value theorem. However, we weren’t able to extend this from infinitesimal rates to chords convincingly, and it turns out this was with good reason: you did not necessarily run a contiguous mile at or above overall pace! However, this is somewhat pathological and unlikely to happen in practice due to how humans realistically pace. Interestingly, the theorem applies widely and does not require aggressive assumptions such as a smooth (differentiable or continuous!) pace. Essentially the only requirement is continuous position (i.e., continuous functions). ↩
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