Definability of Hausdorff Limits for Lipschitz Cells in O-minimal Structures

Abstract

We study Hausdorff limits of definable families over arbitrary models of o-minimal expansions of real closed fields. Over the real field, van den Dries proved that Hausdorff limits of definable families are definable, giving a geometric interpretation of the Marker--Steinhorn theorem. We prove a non-Archimedean analogue for definable families which are Lipschitz cells with a fixed cell presentation and a uniform Lipschitz bound. The proof replaces compactness of closed and bounded subsets of $\mathbb R^n$ by dense completions and long Cauchy sequences, and treats the Hausdorff distance as a metric valued in an ordered completion. We show that Hausdorff limits of such families are standard parts of external fibers over tame extensions, and use stable embeddedness of tame pairs to prove that every such limit is definable in the dense completion of the base model. We also prove a uniform version: the collection of these Hausdorff limits forms a definable family in the dense completion.