  Inspired by the Kantorovich formulation of optimal transport distance between
probability measures on a metric space, Gromov-Wasserstein (GW) distances
comprise a family of metrics on the space of isomorphism classes metric measure
spaces. In previous work, the authors introduced a variant of this construction
which was inspired by the original Monge formulation of optimal transport;
elements of the resulting family are referred to Gromov-Monge (GM) distances.
These GM distances, and related ideas, have since become a subject of interest
from both theoretical and applications-oriented perspectives. In this note, we
establish several theoretical properties of GM distances, focusing on
comparisons between GM and GW distances. In particular, we show that GM and GW
distances are equal for non-atomic mm-spaces. We also consider variants of GM
distance, such as a Monge version of Sturm's $L_p$-transportion distance, and
give precise comparisons to GW distance. Finally, we establish bi-H\"{o}lder
equivalence between GM distance and an isometry-invariant Monge optimal
transport distance between Euclidean metric measure spaces that has been
utilized in shape and image analysis applications.
