Concept

Legendre transform

The construction sending a convex function to the record of its own tangent lines, indexed by slope. It converts minimisation into evaluation, addition into an infimal convolution, and undoes itself — which is why duality appears wherever it does.

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

The chance the average clears 0.75, against the number of draws. The exact probability that the average of n draws exceeds a fixed level, on a logarithmic scale, falling along a straight line whose slope is the rate function, with the normal approximation drawn beside it and diverging.

The tail is not a bell

The limit theorem describes a window of width one over the root of n around the mean; ask instead for the chance that an average lands a fixed distance away and the answer falls exponentially, at a rate computed from the summand before any n is chosen.

probability · Central limit

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.

Central limit theorem Concentration inequality Conjugate Convexity Duality Envelope Heavy tails Involution Large deviations Moment generating function Optimisation Rate function

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