Isoperimetric Problem
Among plane figures with a fixed perimeter, the circle encloses the greatest area.
The isoperimetric problem is one of the oldest optimization problems in geometry. Its visual statement is immediate: deform any closed curve while holding perimeter fixed, and the enclosed area is maximized exactly when the curve becomes a circle.
Formula
Among all simple closed curves of length L enclosing area A, the circle is the unique maximizer of A for given L.
Equality holds if and only if the curve is a circle — proved rigorously by Weierstrass via the calculus of variations.
In Rⁿ, among bodies of given volume, the ball has the least surface area. Equality iff Ω is a ball.
Summary
The isoperimetric problem is one of the oldest optimization problems in geometry. Its visual statement is immediate: deform any closed curve while holding perimeter fixed, and the enclosed area is maximized exactly when the curve becomes a circle.
