Finite-dimensionality of G-spaces

Determine whether every G-space is finite-dimensional in the sense of topological dimension.

Background

A G-space is a complete, locally compact geodesic space with a local unique extension property. The paper identifies as a longstanding problem whether arbitrary G-spaces must have finite topological dimension. This is the first of two conjectures attributed to Busemann.

The paper notes that the finite-dimensionality conjecture is known under additional assumptions, including convexity of metric balls, but remains unresolved in full generality. The main theorem addresses the separate manifold conjecture only for G-spaces whose sufficiently small metric balls are convex, so it does not settle finite-dimensionality for arbitrary G-spaces.