  In analogy with the study of Pollicott-Ruelle resonances on negatively curved
manifolds, we define anisotropic Sobolev spaces that are well-adapted to the
analysis of the geodesic vector field associated with any translation invariant
Finsler metric on the torus $\mathbb{T}^d$. Among several applications of this
functional point of view, we study properties of geodesics that are orthogonal
to two convex subsets of $\mathbb{T}^d$ (i.e. projection of the boundaries of
strictly convex bodies of $\mathbb{R}^d$). Associated with the set of lengths
of such orthogeodesics, we define a geometric Epstein function and prove its
meromorphic continuation. We compute its residues in terms of intrinsic volumes
of the convex sets. We also prove Poisson-type summation formulae relating the
set of lengths of orthogeodesics and the spectrum of magnetic Laplacians.
