Criteria for farthest points on convex surfaces
Jin‐ichi Itoh, Costin Vîlcu · Mathematische Nachrichten · 2009
Abstract We provide a sharp, sufficient condition to decide if a pointyon a convex surfaceSis a farthest point (i.e., is at maximal intrinsic distance from some point) onS, involving a lower boundπon the total curvatureωyaty,ωy≥π. Further consequences are obtained whenωy>π, and sufficient conditions are derived to guarantee that a convex cap contains at least one farthest point. A connection between simple closed quasigeodesicsOofS, pointsy∈S\Owithωy>π, and the set 𝔽 of all farthest points onS, is also investigated (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)