Grothendieck rings of polytopes and non-archimedean semi-algebraic sets

Let be a divisible subgroup of (R, +). Our central result states that, at the level of Grothendieck groups, the classification of -rational polyhedra in Rⁿ up to affine transformations in ⁿ GLₙ (Z) is equivalent to the classification up to affine transformations in ⁿ GLₙ (Q). We prove this by giving an explicit description of these Grothendieck groups. This yields, in particular, a positive answer to the basic case of a question by Hrushovski and Kazhdan; all other cases are still open. As a second application, we give a simple description of the kernel of the motivic volume for non-archimedean semi-algebraic sets, which is a key ingredient of Hrushovski and Kazhdan's theory of motivic integration.