Completeness and constant width in spherical and hyperbolic spaces (Q1911361)

A compact set in a Euclidean (\(E^n\)), spherical (\(S^n\)), or hyperbolic (\(H^n\)) space is called complete if one cannot add a point without increasing its diameter. Following the study of complete sets initiated by \textit{E. Meissner} [Naturforsch. Ges. Zürich 56, 42-50 (1911; JFM 42.0091.01)]) in a Euclidean setting, the author shows examples of complete sets on the sphere and proves that the complete set are the bodies of constant width in \(E^n\) or \(H^n\). Regarding \(S^n\), this characterization is only true for convex sets. Moreover, if the complete convex set on \(S^2\) has diameter \(D>\pi/2\), then it is smooth. The author shows examples of complete non-convex sets on \(S^2\) that do not have constant width.