Existence of Möbius-equivalent antipodal functions for separating functions on general compact Z

Ascertain whether every separating function on a compact metrizable space Z (i.e., a continuous symmetric function ρ: Z×Z → [0, ∞) that vanishes only on the diagonal) admits a Möbius-equivalent antipodal function (i.e., a separating function of diameter one such that for every ξ ∈ Z there exists η ∈ Z with ρ(ξ, η) = 1).

Background

In the finite case, every separating function can be normalized within its Möbius class to an antipodal function, which underpins the authors’ construction and the geodesic structure of the big-Teichmuller space. For general compact metrizable spaces Z, the authors identify the lack of a general result asserting that separating functions admit Möbius-equivalent antipodal representatives as the principal obstacle to proving geodesicity of ((Z), d_Möb).

Resolving this existence question would remove a key barrier in extending results from finite Z to broader classes of compact metrizable spaces and may directly impact the geometric properties (e.g., geodesicity) of the associated big-Teichmuller spaces.