Inequality
Named by 7 essays across 5 fields — each of them below, with the objects they name alongside it.
Choosing what unfair means
Ask whether moving one seat between two regions would make them more equal, and the answer depends on what "equal" is measured in. Three measures, three different answers, and each of the classical methods is the one no transfer can improve for exactly one of them.
A line under every point
The chord above the curve is one definition of convexity. There is a second — a line under the curve at every point, staying under everywhere — and it is the one that turns a statement about a derivative at a point into a statement about the whole function.
The square that cannot be negative
Cauchy–Schwarz is the load-bearing inequality of the whole subject and is nearly always stated without proof. It is one line away from a fact nobody would argue with, and the line is a parabola with no room to cross the axis.
A rectangle cut by a curve
Integration by parts is taught as the product rule run backwards. It is also a picture: an increasing curve cuts a rectangle into two pieces, one of them the area under the curve and the other the area beside it, and the formula says only that the pieces fill the rectangle. Run repeatedly, the same cut produces the factorials and Wallis's product for π.
Every count a solid can have
Euler's formula ties the corners, edges and faces of a convex solid together, but it does not say which numbers are possible. Ernst Steinitz answered in 1906: V corners and F faces occur together exactly when neither is more than twice the other less four. The proof in one direction is two lines of counting; in the other it is two moves — cut off a corner, glue on a tetrahedron — that walk from the pyramids to every allowed pair.
The fewest triangles an edge density allows
Half of all possible edges can be drawn without a single triangle. One more, and triangles appear — not one but several at once. Push the density further and the question becomes a curve: for every share of edges, the fewest triangles a large graph can hold. The answer is a string of scallops, touching a simple parabola at the densities of the balanced multipartite graphs and bulging above it between them. It was guessed in the 1980s and proved in 2008 by a method that turns counting into positive-definite matrices.
Random roots crowd onto the circle
The roots of a random polynomial are not scattered across the plane. They gather within about 1/n of the unit circle, and their directions spread evenly round it. Paul Erdős and Pál Turán proved in 1950 that this is not a fact about randomness at all: any polynomial whose coefficients are all of roughly one size has roots whose directions are nearly even, and the inequality says exactly how nearly, in terms of nothing but the coefficients.
Named alongside it
The objects these essays reach for when they reach for this one.
ConvexityAntiderivativeApportionmentAreaCauchy schwarzCircleComplex numbersConstructionCounting argumentDerivativeDiscriminantDivisor method