### Richard Green

Shared publicly -**Average pace and the Universal Chord Theorem**

If you run a three-mile race at an average pace of six minutes per mile, it will always be the case that you ran for one consecutive mile in exactly six minutes. But if you run a 3.1-mile race at an average pace of six minutes per mile, it does

*not*necessarily follow that you ran a consecutive mile in exactly six minutes.

The key issue here is not the units of measurement, but whether or not the total length of the race is an integer multiple of the length of the subinterval of interest. Assuming for simplicity that this subinterval is one mile, then the following two theorems can be proved, as explained in the recent expository paper

*Average pace and horizontal chords*by

**Keith Burns**,

**Orit Davidovich**and

**Diana Davis**(http://arxiv.org/abs/1507.00871).

**Theorem 1.**If the length of the race is an integer number of miles, then there must be some mile-long interval of the race that was run at exactly the average pace of the whole race.

**Theorem 2.**If, on the other hand, the length of the race is not an integer number of miles, then it is possible that the race was run in such a way that no mile-long interval was run at exactly the overall average pace.

Of course, it is always possible that a mile-long interval of any race is run at exactly the overall average pace; for example, if the entire race were run at a constant speed. What the second result is asserting is that, if the race is not an integer number of miles long, it is possible to find a (smooth) position function for the race so that no mile-long interval is run at the average pace.

The diagram in the bottom right, which comes from the paper, shows an example of the situation covered by Theorem 2. The situation is motivated by two athletic word records: (a) Molly Huddle, who ran a 12km race in a time of 37:49, an average of 3:09 per kilometer; and (b) Mary Keitany, who ran a half marathon (21.1km) in a time of 65:50, an average of 3:07 per kilometer. It is tempting to conclude that Keitany must have run a consecutive 12km stretch of the race at an average pace of 3 minutes and 7 seconds per kilometer, thus beating Huddle's record. However, Theorem 2 shows that this is not necessarily true, because 21.1km is not an integer multiple of 12km. More precisely, the diagram shows a way this could happen: if there is a long, slow stretch in the middle of the race, then every consecutive stretch of 12km will contain the entire slow section of the race and will therefore be run at less than the average pace.

Theorem 1 can be proved with basic calculus, and it was known to the French physicist and mathematician

**André-Marie Ampère**(of electric current fame) as early as 1806. The idea behind the proof is that if a race is n miles long and is subdivided into n consecutive miles, then at least one of these miles was run at greater than equal to the average pace, and at least one of them was run at less than or equal to the average pace. The result then follows by applying the Intermediate Value Theorem.

Theorem 2 was first proved by

**Paul Lévy**in 1934. Lévy considered an equivalent form of the problem, in which the diagram shown is sheared vertically so that the beginning and end of the race both sit on the horizontal axis. A mile-long stretch of the race that is run at average pace then shows up as a horizontal chord of unit length, which partly explains why this result is sometimes known as the

**Universal Chord Theorem**.

The paper also considers some generalizations of Lévy's result, including one proved by

**Heinz Hopf**in 1937. Hopf proved that a set of positive real numbers is the horizontal chord set of a function if and only if the complement of the set is a topologically open set that is additively closed. What Hopf's result means for the athletics scenario is as follows. Given the position function of a race of length L (like the one in the diagram), consider the set S consisting of the lengths of all the distance subintervals of the race that were run at exactly the overall average pace; one of these numbers will be L itself. Hopf's theorem then shows that S is a topologically closed set, meaning that it has an open complement. The theorem also shows that the sum of two lengths that are not in S is also not in S. This forces all the numbers L/n to lie in S whenever n is a positive integer, which is the statement of Theorem 1.

**Relevant links**

Historical mathematicians mentioned in this post:

https://en.wikipedia.org/wiki/André-Marie_Ampère

https://en.wikipedia.org/wiki/Paul_Lévy_(mathematician)

https://en.wikipedia.org/wiki/Heinz_Hopf

Intermediate value theorem: https://en.wikipedia.org/wiki/Intermediate_value_theorem

Photo credit unknown. I found it via Google Images, whose best guess is

*5k runners*.

The

*Lonely Runner Conjecture*is a problem that superficially sounds like the one in this post, but in fact it is much harder. Here are two posts by me about it:

https://plus.google.com/101584889282878921052/posts/YHJkEdKg3jz

https://plus.google.com/101584889282878921052/posts/JMtPiZeVVnT

#mathematics #sciencesunday #spnetwork arXiv:1507.00871

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Richard Green

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+Boris Borcic, the lonely runner conjecture, which I mentioned in the post, is a slightly related result about cyclic motion.

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