We study Minkowski averages on Riemannian manifolds in which the interpolation is by action-minimizing magnetic geodesics with respect to a given magnetic potential. We establish equivalence between Brunn-Minkowski inequalities for this operation and lower bounds on a magnetic Ricci curvature. We then discuss various examples, including natural magnetic fields on Kähler and Sasakian manifolds, and prove a sharp, undistorted Brunn-Minkowski inequality for contact magnetic geodesics on the Heisenberg group. We also observe that closed magnetic potentials from different cohomology classes may give rise to different geodesic Minkowski averages.

32 pages, 4 figures