Isoperimetric inequality

The isoperimetric inequality dates back to the olden days, where there was a problem that asked: “Among all closed curves in the plane of fixed perimeter, which curve (if any) maximises the area of its enclosed region?” This is equivalent to asking: “Among all closed curves in the plane enclosing a fixed area, which curve (if any) minimises the perimeter?” The solution to this problem is expressed in the form of an inequality: the isoperimetric inequality.

The isoperimetric inequality is a geometric inequality that involves the square of the circumference of a closed curve in a plane and the area that it encloses in that plane. This inequality states that

4\pi A\leq L^{2},

where L is the length of a closed curve and A is the area of the planar region that it encloses.

Many proofs have been published for this inequality. For example, in 1938 E. Schmidt produced an elegant proof based on “the comparison of a smooth simple closed curve with an appropriate circle”.

An extension of this inequality is the isoperimetric quotient, Q, of a closed curve which is defined in the following way:

Q={\frac  {4\pi A}{L^{2}}}

Hence, it is the ratio of its area and that of a circle with the same perimeter.

The inequality highlights how  Q ≤ 1 and for a circle Q = 1.

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