How much land does a man need?

Bipolarity

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

micromass

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.

HallsofIvy

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.

nonequilibrium

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)

Bipolarity

Why would it change the result?

BiP

nonequilibrium

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.

Bipolarity

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

Office_Shredder

This is where the problem is not ill-defined. How certain does he become of his velocity? If with 1 minutes left he can go 'oh shoot I'm slower than I thought, better close up the loop now" why couldn't he do that 10 seconds into the race and re-adjust the route to make it a smaller circle. The details of how well he knows his speed and his position at each point in time need to be fleshed out

nonequilibrium

I don't see a real problem: the probability distribution (or maybe a better word is ignorance distribution if we presume that his velocity is constant but simply unknown; another option is taking a genuine probability distribution, like brownian motion) of his velocity is constant in time, but at each moment in time he recalculates the optimal path based on both this given distribution and the remaining time. Am I overlooking something?

avarga

In the story, the land gets more valuable as he gets further from his starting point.
So let's assume he knows his speed, can travel it consistently, and travels 1 unit in a day.
Now if the value of the land is given by v(x,y) = 1, we agree a circle is the best path.
But what if v(x,y) = y? or v(x,y) = sqrt(x^2+y^2)?

avarga

I found an approximation of the optimized path when the land value increases linearly as you go east. Finding an exact function won't be so easy.

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