Tropical convexity for nonquadratic tropical q-Fréchet means

Determine whether tropical q-Fréchet means for positive powers q other than 1 and 2 exhibit tropical convexity in the sense of tropically convex location problems, and establish whether they can be analyzed within the same framework as the q=1 and q=2 cases.

Background

The paper places the q=1 and q=2 cases within the class of tropically convex location problems. This framework provides structural results for the tropical Fermat–Weber problem and for squared-distance tropical Fréchet means.

It remains unresolved whether the other positive-power cases share this tropical convexity property and whether the same theoretical framework can be applied to them. The question concerns the extension of the paper’s geometric and optimization results beyond the two established exponents.

References

The cases $q=1,2$ constitute tropically convex location problems in the sense of \citeauthor{Comǎneci:2024} , which then opens the question of whether the remaining cases also exhibit tropical convexity in the same sense and whether they may be studied in the same framework.

— Tropical Fréchet Means  (2502.05322 - Lin et al., 7 Feb 2025) in Section Discussion