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

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.