Explore a 24-minute lecture on generalizing the Minkowski average of two subsets in Riemannian manifolds, replacing geodesics with arbitrary families of parametrized curves. Delve into the characterization of curve families on Riemannian surfaces that satisfy Brunn-Minkowski inequalities for given volume forms. Examine the conditions under which constant-speed curves on Riemannian surfaces fulfill Brunn-Minkowski with respect to the Riemannian area form, focusing on the relationship between geodesic curvature, Gauss curvature, and a specific inequality involving these elements.