On the Structure of Busemann Spaces with Non-Negative Curvature

TL;DR

This paper extends the structure theory of Alexandrov spaces to Busemann spaces with non-negative curvature, establishing rectifiability, measure properties, and tangent cone uniqueness.

Contribution

It introduces new structural results for Busemann spaces with non-negative curvature, including measure and rectifiability properties, and links to Finslerian spaces.

Findings

Spaces are rectifiable with unique tangent cones almost everywhere.

Existence of a non-trivial Hausdorff measure under certain conditions.

Refined estimates for Hausdorff dimension of singular strata.

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 integer-dimensional Hausdorff measure, and satisfies the measure contraction property. We also show that such spaces are rectifiable and that almost every point admits a unique tangent cone isometric to a finite-dimensional Banach space. In addition, under mild control of the uniform smoothness constant, we obtain refined estimates for the Hausdorff dimension of the singular strata. Our results not only enrich the theory of synthetic sectional curvature lower bound for metric spaces, but also provide some useful tools and examples to study Finslerian metric measure spaces.