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 ω y at y , ω y ≥ π . Further consequences are obtained when ω y > 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 ω y > π , and the set of all farthest points on S , is also investigated (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)