isoperimetric inequality

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If p is the perimeter of a closed curve in a plane and the area enclosed by the curve is A, then p2≤4πA, and equality is only achieved if the curve is a circle. Turning this condition round the other way, it says that for a fixed length of curve p the greatest area which can be enclosed is when the curve is a circle. The result can be generalized to surfaces in 3-dimensional space where the sphere is the most efficient shape at enclosing volume for a fixed surface area, and to higher dimensions.

Subjects: Mathematics.

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