Concept

Supporting line

A line touching a curve at a point and lying below it everywhere. A convex function has one at every point, which is what turns a statement about a derivative at one place into a bound holding across the whole domain.

Named by 3 essays across one field — 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

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.

analysis · Convexity

x² and its conjugate. Two panels: the curve x² with tangent lines of several slopes, and the conjugate function plotted against slope, whose value at each slope is the intercept of the corresponding tangent.

The function seen from its tangents

A convex function is the upper envelope of its own tangent lines, so it can be described by giving, for each slope, how far the line of that slope has to be pushed down. That description is a second function, and applying the construction twice returns the original.

analysis · Convexity

Named alongside it

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

Convexity Optimisation Arithmetic mean Centroid Chord Conjugate Derivative Duality Envelope Geometric mean Gradient Inequality

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