Isoperimetric inequality

The isoperimetric inequality states that a plane region with perimeter P and area A satisfies P² ≥ 4πA, with equality for a circle.

Isoperimetric deficit: The difference between a region’s perimeter-based upper bound on area and its actual area. It quantifies how far a region falls short of the circle’s maximal area.
Dido's problem: The classical problem of enclosing the greatest area with a fixed length of boundary. Its legendary formulation expresses the geometric question behind the inequality.
Cheeger's inequality: A bound relating the first nonzero eigenvalue of a manifold’s Laplacian to its Cheeger constant. Isoperimetric bounds help connect small boundaries to low-frequency behavior.
Spherical isoperimetric inequality: The inequality bounding the area enclosed by a curve on a sphere in terms of its length. Positive curvature changes the sharp area bound from the planar formula.
Isoperimetric ratio: A dimensionless ratio comparing a shape’s area with the area of a circle having the same perimeter. It turns the inequality into a scale-independent measure of shape efficiency.
Zenodorus: An ancient Greek mathematician known for work on geometric figures of equal perimeter and area. His surviving results include early comparisons of areas and perimeters.
Isoperimetric problem: The problem of optimizing a geometric quantity under a constraint on another quantity. The plane inequality is the classical solution for maximizing area at fixed perimeter.
Higher-dimensional isoperimetric inequality: The inequality relating boundary measure and volume for regions in dimensions above two. It generalizes the planar result, with spheres replacing circles as equality cases.
Circle: The plane curve consisting of all points at a fixed distance from a center. It is the unique plane shape attaining equality for fixed perimeter.
Jakob Steiner: A nineteenth-century Swiss mathematician who developed influential methods in geometry. His geometric arguments helped establish the circle’s optimality, though required an existence argument.

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