Packing and doubling in metric spaces with curvature bounded above

We study locally compact, locally geodesically complete, locally CAT(κ) spaces (GCBA κ-spaces). We prove a Croke-type local volume estimate only depending on the dimension of these spaces. We show that a local doubling condition, with respect to the natural measure, implies pure-dimensionality. Then we consider GCBA κ-spaces satisfying a uniform packing condition at some fixed scale r or a doubling condition at arbitrarily small scale, and prove several compactness results with respect to pointed Gromov–Hausdorff convergence. Finally, as a particular case, we study convergence and stability of Mκ-complexes with bounded geometry.