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Concept

Inequality

A statement that one quantity is at least another. Most named ones are a single supporting line at a well-chosen point of a convex function, which is why collecting convex functions is a way of collecting inequalities.

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

What each rule is answering. A table of five apportionments against three measures of inequality between two regions, with a tick where no transfer of a seat reduces the measure; each measure certifies exactly one of the five.

Choosing what unfair means

Ask whether moving one seat between two regions would make them more equal, and the answer depends on what "equal" is measured in. Three measures, three different answers, and each of the classical methods is the one no transfer can improve for exactly one of them.

applied · Apportionment A line under every point of x². The curve x² with 4 tangent lines drawn, each extended across the whole interval and each staying below the curve throughout.

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

Named alongside it

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

ApportionmentConvexityCounting argumentDerivativeDivisor methodGeometric meanGradientLocal minimumLocal optimumOptimisationQuotaSupporting line

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