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- Why Circle encloses largest Area? 8 answers
An old story I heard starts by two people that was arguing about how much land a man need. So they called to a young man, and said to him:
You are stating in that point, and start running all the day. All the area you cover before sunset will be yours. But you must return to the original point, otherwise you get nothing
So I considered what is the best route for him to get the maximal area.
More mathematical form:
Given a number d, what is the continues closed shape with the maximal area and perimeter d?
If we are limited to rectangles, the answer is "a square", what leads me to guess the optimal shape is a circle. Is it true and can it be proven?
What I can think about is that the shape must be convex, otherwise we can 'cut' it and save line. But I have no idea how to eliminate some strange elliptic shapes.