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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.

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    For completeness, here's the link to our paper with a proof of the Euclidean cone case: http://naml.us/~irving/papers/irving_segerman2012_fractal.pdf2012-12-05

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