Convexity has many applications in everyday life. One of the most elementary settings in which it arises in economics is consumer theory. In our world, a vast array of goods and services is available, so making a selection can sometimes be difficult. Each consumer chooses differently, according to their wants and preferences, which must be determined more or less rationally. But wants are not the only consideration: consumption choices are also shaped by certain constraints. Products are not available in unlimited quantities and, above all, every consumer has a necessarily limited budget b. These observations provide the basis for a mathematical model of the problem.
Consumer baskets ---------------------------------
Suppose that exactly n goods and services are available on the market. Each is available in limited quantities, giving rise to a subset X of ℝ*n representing the collection of consumer baskets. Each basket is represented by a vector x = (x*1, x2… *xn*), subject, of course, to the available quantity of each good.
This set is naturally convex. Indeed, for any pair of consumer baskets satisfying the availability constraints, every weighted linear combination of the two baskets—with positive coefficients summing to 1—will also satisfy them. That is precisely the definition of convexity!
In a convex set E, the barycenter of any number of points in E with positive coefficients also belongs to E. Its coordinates are obtained by taking a "weighted" linear combination of the coordinates of these points (which are elements of ℝ*n*), then dividing the result by the sum of the coefficients so that they sum to 1.