Full non-Archimedean Hausdorff-limit theorem and geometric extension

Establish the full non-Archimedean analogue of the definability theorem for Hausdorff limits beyond bounded Lipschitz-cell families, together with a corresponding extension of the Lion–Speissegger geometric proof.

Background

The paper proves definability of Hausdorff limits only for bounded definable families whose fibers are Lipschitz cells with a fixed cell presentation and a uniform Lipschitz bound. This restriction is essential because a finite cell decomposition of a Hausdorff-Cauchy family need not yield Hausdorff-Cauchy families on the individual cells.

The unresolved problem is to remove the Lipschitz-cell hypothesis and obtain the full non-Archimedean counterpart of the real-field Hausdorff-limit definability theorem. The authors also identify as future work the development of a corresponding extension of the Lion–Speissegger geometric proof, which in the real-field setting uses geometric constructions involving tangent spaces, Grassmannians, blow-ups, and integrable distributions.