2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/138993A spherical set is called convex if for every pair of its points there is at least one minimal geodesic segment that joins these points and lies in the set. We prove that for n >= 3 a complete locally-convex (topological) immersion of a connected (n-1)-manifold into the n-sphere is a surjection onto the boundary of a convex set.15 pages, 3 figures. Two more pictures. Corrections, mostly notational have been made. Proofs are given in more detailMetric GeometryDifferential Geometry53C45text