Entropy-Driven Geometry in Non-Reflexive Banach Spaces: Metric Constructions, Curvature Bounds, and Machine Learning Applications

This paper develops a comprehensive framework for geometric analysis in non-reflexive Banach spaces through the introduction of novel intrinsic metrics and their applications to machine learning. We first construct entropy-driven metrics that induce topologies strictly finer than weak-∗ topologies while preserving completeness, and establish curvature lower bounds in variable-exponent spaces extending optimal transport theory. Our main results demonstrate how these geometric structures enable: (1) linear convergence of gradient flows to sharp minima despite the absence of Radon-Nikody´m property, (2) non-Euclidean adversarial robustness certificates for deep neural networks, and (3) sublinear regret bounds in sparse optimization via Finsler geometric methods. A fundamental non-reflexive Nash embedding theorem is proved, revealing obstructions to reflexive space embeddings through entropy distortion. The theory is applied to derive approximation rates in variable-exponent spaces and accelerated optimization in uniformly convex entropy-augmented norms. These results bridge functional analytic geometry with machine learning, providing new tools for non-smooth optimization and high-dimensional data analysis.