Local metric properties and regular stratifications of $p$-adic definable sets

Abstract

We study the geometry of germs of definable (semialgebraic or subanalytic) sets over a $p$-adic field from the metric, differential and measure geometric point of view. We prove that the local density of such sets at each of their points does exist. We then introduce the notion of distinguished tangent cone with respect to some open subgroup with finite index in the multiplicative group of our field and show, as it is the case in the real setting, that, up to some multiplicities, the local density may be computed on this distinguished tangent cone. We also prove that these distinguished tangent cones stabilize for small enough subgroups. We finally obtain the $p$-adic counterpart of the Cauchy–Crofton formula for the density. To prove these results we use the Lipschitz decomposition of definable $p$-adic sets of [5] and prove here the genericity of the regularity conditions for stratification such as $(w_f)$, $(w)$, $(a_f)$, $(b)$ and $(a)$ conditions.