Concept

Geometric mean

The nth root of the product of n positive numbers, which is the average of their logarithms exponentiated. It is never larger than the arithmetic mean, with equality only when every number is the same, and the gap between the two is the logarithm's concavity.

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

Two points, and everything one round of compass and straightedge adds. Two starting points with the line and circles they permit, and the four points where those objects cross.

What two points can build

A compass and a straightedge are not a craft. They are two operations on a set of points, applied over and over, and writing them that way turns "can this be drawn?" into a question with an answer.

computation · constructible numbers

A chord of x², and the curve under it. The curve x² with one chord drawn across it, the region between them shaded, and the midpoint heights of both marked. The comparison is computed at four hundred sample points.

The curve of the average, and the average of the curve

A curve that bends upwards keeps every one of its chords above it. That single fact, applied to a weighted average instead of a midpoint, turns into an inequality that produces the arithmetic–geometric mean inequality, Cauchy–Schwarz and the entropy bound as special cases.

analysis · convexity

Named alongside it

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

Arithmetic mean Centroid Chord Circle Closure Constructible number Convexity Field extension Incidence Jensen inequality Operation set Second derivative

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