Where does convexity collide?

In geometry, convexity is the property of a set that contains, for every two of its points, the whole straight segment joining them, so a convex body is a solid shape with no dents, notches, or holes: a ball, a cube, and a lens are convex, while a star and a crescent are not. Schneider's 2013 Convex Bodies: The Brunn-Minkowski Theory is the standard modern treatment of such bodies and of their volumes, surfaces, and mixed measures. Here the referent is a shape sitting in space, described by its boundary and volume, not a curving cost function to be minimised, a bond's price curvature, or an assumption about a shopper's tastes.[1]

Unit

A convex set or body in space, given by its points and its boundary

Recovery means

Not applicable

Measured by

Characterised by volume, surface area, width, and mixed volumes, and built as the convex hull of a set of points

In optimization and analysis, convexity is the property of a function whose graph cups upward, so the straight line between any two points on the graph never dips below it: this is exactly the condition, stated formally by Jensen's inequality, that makes any local low point a global minimum and lets a problem be solved reliably. Rockafellar's 1970 Convex Analysis is the founding systematic account of convex functions and sets and the calculus of subgradients and duality built on them. Here the referent is an algebraic property of a function or feasible region that guarantees tractable minimisation, not a solid shape studied for its volume, a bond's price curvature, or a preference axiom.[2]

Unit

A convex function or convex feasible set, as a mathematical property

Recovery means

Not applicable

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Checked from the definition or a nonnegative second derivative (a curvature matrix that is never negative), and exploited by convex optimisation algorithms

In fixed-income finance, convexity is the curvature of the relationship between a bond's price and its yield: duration gives the straight-line, first-order guess at how the price moves when yields change, and convexity is the second-order correction that captures how that price sensitivity itself changes, so a bond with high positive convexity gains more when yields fall than it loses when they rise. Dunetz and Mahoney's 1988 study Using Duration and Convexity in the Analysis of Callable Bonds showed how the measure is applied and how an embedded call option can drive convexity negative. Here the referent is a number pinned to one specific price-yield curve, quoted alongside duration, not a shape in space, a general property of a cost function, or a claim about consumer tastes.[3]

Unit

One bond's price-yield curvature, a number quoted with its duration

Recovery means

Not applicable

Measured by

Computed as the second derivative of price with respect to yield, scaled by price, from the bond's cash flows

In microeconomics, convexity is an assumption placed on a person's preferences: if a shopper is content to accept either of two different bundles of goods, then a blend of the two, an average, is at least as good as either extreme, which yields smooth, bowed-in indifference curves and a taste for variety. Arrow and Debreu's 1954 Existence of an Equilibrium for a Competitive Economy made convexity of preferences and of production sets a load-bearing assumption in the proof that a competitive market has an equilibrium. Here the referent is a behavioural axiom about how choices rank, tested against observed demand, not a solid shape, a property of a cost function, or a bond's price curvature.[4]

Unit

An axiom on a preference ordering: an average weakly beats the extremes

Recovery means

Not applicable

Measured by

Expressed as convex upper-contour sets, or bowed-in indifference curves, and probed with revealed-preference and demand data