Cauchy schwarz
Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.
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.
One subtraction clears a direction
A basis is a set of directions to measure along, and most bases are awkward because the directions get in each other's way. Removing one shadow at a time turns any basis into one where every coordinate is a shadow and nothing interferes.
A plane disguised as an arrow
The cross product of two arrows is an arrow perpendicular to both, as long as the area of their parallelogram. Reflect everything in a mirror and it points the wrong way, because it was never an arrow: it is a plane, written as the one direction a plane in three dimensions leaves over.
The formula that cannot share
A circuit computes the parity of n letters with a chain of n − 1 exclusive-or gates. A formula in and, or and not, which may use each result only once, needs n² letters at least — a bound Khrapchenko proved by counting the cube's edges between true and false — and exhaustive search over every formula of four letters finds parity sitting exactly on it.
No graph is poorer in squares than chance
Take any graph with a fixed share of its possible edges and count how often four points chosen in order close up into a ring. However the edges are arranged — in cliques, in two-sided blocks, at random — the count never falls below what a random graph of the same density gives. Sidorenko conjectured that the same holds for every pattern without an odd cycle. For a triangle it is false; for a path or a ring of four it takes two lines; for K₅,₅ with a ten-cycle removed nobody knows.
Named alongside it
The objects these essays reach for when they reach for this one.
Inner productOrthogonalityDot productPositive definiteProjectionApproximationBasisBipartite graphBoolean formulaCircuit complexityCounterexampleCross product