Banach nature of tangent cones at almost every point in Busemann concave spaces

Ascertain whether almost every point in a Busemann concave metric space admits a tangent cone that is isometric to a finite-dimensional Banach space; equivalently, prove or refute that tangent cones are Banach spaces for H^n-almost all points under natural hypotheses on the space.

Background

The authors highlight that Busemann concavity is unstable under Gromov–Hausdorff limits, complicating analysis of tangent cones. While prior results show uniqueness of tangent cones under additional assumptions and in certain cases identify Carnot group structures, it remains unclear in general whether tangent cones are Banach spaces almost everywhere.

Clarifying the Banach nature of tangent cones in the broad class of Busemann concave spaces would strengthen the infinitesimal geometric framework and align it with the refined structure theory established in Alexandrov geometry.