Concept

Weighted average

A sum of values each multiplied by a non-negative weight, with the weights adding to one. The result always lies between the smallest and largest value, and in the plane any weighted average of points lies inside their convex hull.

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

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

The roots of the derivative inside the hull of the roots. Three polynomials of degree five, each drawn as its roots with their convex hull shaded, and the four roots of its derivative marked. Every root of the derivative lies inside the hull.

The roots of the slope stay inside

Mark the roots of a polynomial in the complex plane and stretch a band around them. However the roots are arranged, the roots of the derivative land inside the band — never outside, never on a new frontier. The reason is a balance of pushes, the same reason makes the derivative of a cubic mark the foci of an ellipse nobody asked for, and a question about how far inside the roots must sit has been open since 1958.

algebra · Polynomial roots

Named alongside it

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

Centroid Arithmetic mean Chord Complex numbers Convex hull Convexity Derivative Ellipse Focus Geometric mean Jensen inequality Polynomial

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