Self-Generated Measures and the Centroid Rigidity of Power Laws

TL;DR

This paper reveals a rigidity property of power law functions through a classical calculus problem involving the centroid of a function's subgraph, linking geometric scaling to power law characterization.

Contribution

It proves that the geometric scaling property uniquely characterizes power law functions among twice differentiable positive functions.

Findings

Power laws satisfy the geometric scaling property with a specific constant.

The proof employs a probabilistic approach using a self-generated measure.

The characterization extends to functions with locally Lipschitz elasticity.

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.