Self-Generated Measures and the Centroid Rigidity of Power Laws

Abstract

We revisit a classical calculus computation: the centroid of the subgraph of a function on the interval from 0 to a, and show that it hides a rigidity theorem. Let f be twice continuously differentiable on (0, infinity), take values in (0, infinity), and satisfy f(0+) = 0. Define xbar(a) as (integral from 0 to a of x f(x) dx) divided by (integral from 0 to a of f(x) dx), and define ybar(a) as (1/2) times (integral from 0 to a of f(x)^2 dx) divided by (integral from 0 to a of f(x) dx). We prove that the Geometric Scaling Property, namely the identity ybar(a) = lambda * f(xbar(a)) for every a > 0, holds if and only if f(x) = A * x^p with A > 0 and p > 0. For these power laws the optimal constant is lambda = (p+1)/(2(2p+1)) * ((p+2)/(p+1))^p. After a scale-free normalization, the proof is probabilistic: with the self-generated probability measure on (0, a) having density proportional to f, we have xbar(a) equal to the expected value of Xa and ybar(a) equal to (1/2) times the expected value of f(Xa), so the Geometric Scaling Property becomes an equality in expectation across all truncation scales. Differentiating with respect to a yields a weighted mean identity for the elasticity E(x) = x f'(x) / f(x); a second differentiation forces a vanishing variance principle that makes E constant, hence f a pure power, and the stated value of lambda follows. The argument uses no asymptotics and extends to f that is once continuously differentiable on (0, infinity) with locally Lipschitz elasticity.