Isoperimetric Inequality

A family of geometric inequalities that bound the boundary measure (perimeter, surface area) of a region from below in terms of its volume, asserting that among sets with fixed volume the ball (sphere) minimizes the boundary measure.

The statement in planar Euclidean geometry that for any bounded region with perimeter P and area A we have 4πA ≤ P², with equality exactly for round disks; equivalently, among plane regions with given perimeter the circle encloses the maximal area.