# How much land does a man need?

by Bipolarity
Tags: land
 P: 783 So the story by Leo Tolstoy goes, that a man has a limited amount to time to run a closed loop in an open field and when the time is up, he will get all the land that he has enclosed. If he cannot run an enclosed loop, then he will get nothing. Suppose that the man has a fixed speed that he runs at, and which he has already calculated. What shape could he run in so that he could get the most land possible? I would imagine a circle, but there are infinite possible shapes with all sorts of twists and turns so I can't prove my answer, but was wondering if someone here knew the proof or even whether a solution actually exists? What branch of mathematics would this fall under? Calculus of variations? Circle sounds quite elegant though. BiP
 Mentor P: 18,346 This would be the isoperimetric problem. The circle is indeed the correct answer. There are many proofs of this theorem. One can indeed use calculus on variations if the curve he runs is sufficiently smooth. The theory of Fourier series also is applicable to the problem. See http://en.wikipedia.org/wiki/Isoperimetric_inequality See http://www.math.utah.edu/~treiberg/isoperim/isop.pdf for many nice proofs of the problem.
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,682 This reminds me of the "Running Grant". In early North American colonial days, settlers negotiated with a local American Indian tribe for the property along a river in the Pennsylvania colony. The land was to be from the river to the hills and as far along the river as a man could run in one day. The European settlers "chose" to interpret "a day" as 24 hours, graded a path along the river, brought in a trained runner, and had runners with torches for short stretches during the night. Of course, later, it turned out that it was another tribe that claimed that land.
 P: 1,422 How much land does a man need? I wonder what the optimal shape would be depending on his certitude about how fast he runs... (the circle being the case for the absolute certitude)
P: 783
 Quote by mr. vodka I wonder what the optimal shape would be depending on his certitude about how fast he runs... (the circle being the case for the absolute certitude)
Why would it change the result?

BiP
 P: 1,422 Because if you're not exactly sure about how fast you run, you can't know how much distance you can traverse in the allotted time, hence it's no longer an isoperimetric problem. Intuitively, you would like to allow the possibility of closing the loop earlier than expected (if it turns out time is running out), so maybe (again, intuitively) the path would be flatter. But anyway, I'm sorry for the irrelevant excursion, it isn't what you asked.
 P: 783 It's ok, the original problem is closely related. So let's suppose that the speed of the runner ranges anywhere from 0 to x where any value between 0 and x has uniform probability. Let's assume that this speed remains the same throughout the problem but the runner can never be exactly sure what it equals. If the runner assumes that his/her speed is x/2, then he will complete the loop with 50% probability but the loop will be quite large. If the runner gambles for larger speed, the loop will be larger but the probability that he will complete the loop is reduced. Then the ultimate measure of the technique should be the size of the loop multiplied by the probability that he completes the loop? Or can we find a better measure of the technique's effectiveness? One thing to realize is that the runner will try to complete the loop if he sees that time is not enough. In other words, he would become certain of his velocity as he runs which greatly complicates the problem. BiP
Emeritus