On the Structure of Busemann Spaces with Non-Negative Curvature I

Abstract

We extend the structure theory of Burago--Gromov--Perelman for Alexandrov spaces with curvature bounded below, to the setting of Busemann spaces with non-negative curvature. We prove that any finite-dimensional Busemann space with non-negative curvature, satisfying Ohta's $S$-concavity and local semi-convexity, admits a non-trivial Hausdorff measure of integer dimension, satisfies the measure contraction property and contains an open dense topological manifold part with full measure. We also show that such spaces are rectifiable and almost every point admits a unique tangent cone isometric to a finite-dimensional Banach space. Finally, based on a refined analysis on strainers and asymmetric angles, we obtain a Hausdorff dimension estimate for singular strata, and show that such spaces admit a natural measure-theoretic stratification. Our results enrich the theory of synthetic sectional curvature lower bounds for metric spaces, and provide some useful tools and examples to study Finslerian metric measure spaces.

arXiv comment43 pages; minor corrections and filling the gap of the proofs of Theorem 7.6 and 7.17 according to reviewers' suggestions; accepted by Trans. Amer. Math. Soc