  We prove the nonarchimedean counterpart of a real inequality involving the
metric entropy and measure geometric invariants $V_i$, called Vitushkin's
variations. Our inequality is based on a new convenient partial preorder on the
set of constructible motivic functions, extending the one considered by R.
Cluckers and F. Loeser in Constructible motivic functions and motivic
integration, Invent. Math., 173 (2008). We introduce, using motivic integration
theory and the notion of riso-triviality, nonarchimedean substitutes of the
Vitushkin variations $V_i$, and in particular of the number $V_0$ of connected
components. We also prove the nonarchimedean global Cauchy-Crofton formula for
definable sets of dimension $d$, relating $V_d$ and the motivic measure in
dimension $d$.
