An Information-Geometric Program for AM-GM Stability: Log-Sobolev Bounds and Variance Control

This work proposes and develops an information-geometric program to establish a quantitative stability bound for the Arithmetic Mean– Geometric Mean (AM-GM) inequality De Ceuster [2025b]. We combine tools from information geometry, optimal transport, and functional inequalities (specifically the Log-Sobolev inequality, LSI). The core philosophy is to translate additive dispersion (variance σ2) into multiplicative deviation (log(A/G)) via probabilistic metrics. We formulate a sufficient-condition theorem: if an empirical distribution satisfies an LSI, an explicit exponential bound for the AM/GM ratio follows.