In this paper we give a complete characterisation of linear repetitivity for cut and project schemes with convex polytopal windows satisfying a weak homogeneity condition. This answers a question of Lagarias and Pleasants from the 90s for a natural class of cut and project schemes which is large enough to cover almost all such polytopal schemes which are of interest in the literature. We show that a cut and project scheme in this class has linear repetitivity exactly when it has the lowest possible patch complexity and satisfies a Diophantine condition. Finding the correct Diophantine condition is a major part of the work. To this end we develop a theory, initiated by Forrest, Hunton and Kellendonk, of decomposing polytopal cut and project schemes to factors. We also demonstrate our main theorem on a wide variety of examples, covering all classical examples of canonical cut and project schemes, such as Penrose and Ammann–Beenker tilings.