Given a Riemannian surface with nonnegative Gaussian curvature, the area of a ball of radius $r$ around any point has area at most $\pi r^2$. I have a simple proof of this in the Euclidean cone case (a surface which is flat except at a discrete set of cone points), which shows that all points in the ball can be reached with geodesics that do not pass through a cone point.
My question: who would have proved this first, and does anyone know a good reference for such? All of my searches find much more complicated results involving sectional curvature in higher dimensional manifolds.