Relaxed Triangle Inequality for Rényi Divergence of Order 1/2 Between Gaussian Distributions

Superseded. This record is an earlier version and is kept only for the record. The current version of this work is Sharp Relaxed Triangle Inequality for Rényi Divergence of Order 1/2 Between Univariate Gaussians. Please cite that record instead. This work initiates the study of relaxed triangle inequalities for Rényi divergence between Gaussian distributions. While the analogous problem for Kullback-Leibler divergence was recently resolved by Xiao, Huo, Yan, and Xu (2026), the Rényi case remained completely open. We focus on the order one-half and the one-dimensional Gaussian family, where the Rényi divergence coincides with the Bhattacharyya distance. Our main contributions are: (i) an explicit reduction of the extremal problem to a finite-dimensional optimization over mean and covariance budgets with a two-branch inversion structure; (ii) an exact closed-form solution under symmetric constraints, obtained via a transcendental equation with a unique critical point; (iii) a rigorous proof that, unlike the KL case, the supremum cannot be achieved when all three distributions share the same mean — revealing a fundamental structural difference between KL and Rényi divergences in the triangle inequality setting. We also provide sharp corner and boundary values, a complete branch-dominance classification, and an explicit first-order optimality system for the general asymmetric case, which remains open.