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

Let $\mathrm{\Gamma }$ be a divisible subgroup of $(\mathbb{R},+)$. Our central result states that, at the level of Grothendieck groups, the classification of $\mathrm{\Gamma }$-rational polyhedra in ${\mathbb{R}}^n$ up to affine transformations in ${\mathrm{\Gamma }}^n⋊{\mathrm{G}\mathrm{L}}_n(\mathbb{Z})$ is equivalent to the classification up to affine transformations in ${\mathrm{\Gamma }}^n⋊{\mathrm{G}\mathrm{L}}_n(\mathbb{Q})$. 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.