Criteria for farthest points on convex surfaces

We provide a sharp, sufficient condition to decide if a point y on a convex surface S is a farthest point (i.e., is at maximal intrinsic distance from some point) on S, involving a lower bound π on the total curvature ωy at y, ωy ≥ π. Further consequences are obtained when ωy > π, and sufficient conditions are derived to guarantee that a convex cap contains at least one farthest point. A connection between simple closed quasigeodesics O of S, points y ∈ S\O with ωy > π, and the set of all farthest points on S, is also investigated (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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