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Concept

Isoperimetric inequality

The statement that a closed curve of length L encloses at most L²/4π of area, with equality only for the circle. Its relatives say the same of other quantities — the lowest eigenvalue of a region among them — and symmetrisation proves most of them.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

How much of a sphere lies near its equator. Curves of the share of the sphere within ε of the equator against ε, for spheres in 3, 10, 100, 1000 dimensions: a straight line in three dimensions, a near step in a thousand.

A sphere that is nearly all equator

On an ordinary globe, the band within a tenth of the radius of the equator holds a tenth of the surface. On a sphere in a thousand dimensions the same band holds 99.85% of it, and the band of a fifth holds all but about two parts in ten billion. Almost every point of a high-dimensional sphere is near every equator at once — and so any function that cannot change quickly is, over almost all of the sphere, almost constant.

probability · Concentration The lowest mode of the regular pentagon. Level lines of the first Dirichlet eigenfunction of a regular 5-gon; λ times area 18.9191.

Three sides and four are proved

Among all shapes of a given area, the disc has the smallest lowest eigenvalue. Among triangles it is the equilateral one, among quadrilaterals the square — both proved by sliding chords to an axis. For five sides the regular pentagon wins every computation, every nudge raises its value by the square of the nudge, and there is still no proof.

geometry · Isoperimetric

Named alongside it

The objects these essays reach for when they reach for this one.

Concentration of measureCurse of dimensionalityDimensionEigenvalueLaplacianLipschitz functionLocal minimumNormal distributionOrthogonalityRegular polygonSphereSymmetrisation

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