  Convex curvature properties are important in designing and analyzing convex
optimization algorithms in the Hilbertian or Riemannian settings. In the case
of the Hilbertian setting, strongly convex sets are well studied. Herein, we
propose various definitions of strong convexity for uniquely geodesic sets in a
Riemannian manifold. We study their relationship, propose tools to determine
the geodesic strongly convex nature of sets, and analyze the convergence of
optimization algorithms over those sets. In particular, we demonstrate that the
Riemannian Frank-Wolfe algorithm enjoys a global linear convergence rate when
the Riemannian scaling inequalities hold.
