Lipschitz function
Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
A cut that gets sharper as the map gets lazier
Iterating a map that shrinks distances by 0.9999 takes 141,254 steps to reach its fixed point to a millionth. Cutting takes six: read the map at one point, and the fixed point must lie on the far side of the line halfway between the point and its image. The cut gets better exactly where iteration gets worse, because a map that barely shrinks puts its fixed point almost on every such line.
Named alongside it
The objects these essays reach for when they reach for this one.
AlgorithmBisectionConcentration of measureContraction mappingConvexityCurse of dimensionalityDimensionFixed pointIsoperimetric inequalityNormal distributionOrthogonalitySphere