Extension of the Alberti–Marchese structure theorem beyond finite-dimensional Euclidean spaces

Ascertain whether the Alberti–Marchese structure theorem (Theorem 1.1 in Alberti–Marchese), which represents a classical k-dimensional flat chain in ℝ^n as the restriction of a boundaryless normal k-current with controlled mass, extends to ambient spaces beyond finite-dimensional Euclidean space (for example, to Banach spaces) in dimensions k ≥ 2.

Background

The proof strategy for representing currents as restrictions of boundaryless normal currents relies on a strict polyhedral approximation theorem. This paper establishes such an approximation for k=1 in spaces with a conical geodesic bicombing, enabling a Banach-space extension in dimension 1.

For k≥2, the strict polyhedral approximation fails in infinite-dimensional Hilbert spaces, and thus current techniques do not directly yield an extension of the Alberti–Marchese theorem beyond finite-dimensional Euclidean settings. Whether an analogue holds more generally remains unresolved.