Sub-Finslerian Interpolation Inequalities

In this paper, we prove that forward ideal sub-Finslerian manifolds support interpolation inequalities for optimal transport, extending the results of Barilari and Rizzi, arXiv:1705.05380, from the sub-Riemannian to the sub-Finslerian setting. A key role is played by the introduction of sub-Finslerian Jacobi fields and the establishment of optimal transport theory on sub-Finslerian manifolds. By combining this transport framework with sub-Finslerian Jacobian estimates, we characterize the generalized distortion coefficients. As an application, we deduce several fundamental geometric inequalities, including the Brunn-Minkowski and Borell-Brascamp-Lieb inequalities. Finally, for the case of the Randers sub-Finslerian Heisenberg group, whose metric is defined by a sub-Riemannian metric perturbed by a drift term, we explicitly show that it satisfies the measure contraction property.