Geodesics and their geometry in Regimes III–V

Determine whether geodesics exist in Regimes III–V and characterize the minimum hop count \(H_{\lceil n x\rceil}(T_n)\) among geodesics, together with the longest-edge length and, in dimensions \(d\ge2\), the transverse fluctuations of geodesics.

Background

Outside the regimes where finite balls and geodesic attainment are established under additional conditions, the paper does not determine whether passage-time minimizers exist. Even when geodesics exist, the constructed upper-bound paths need not represent their geometry.

The authors explicitly raise unresolved questions about geodesic hop counts, the length of the longest edge, and transverse fluctuations in the higher-dimensional model.