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In this paper, we first establish the separation theorem between a point anda locally geodesic convex set and then prove the existence of a supportingquasi-hyperplane at any point on the boundary of the closed locally geodesicconvex set on a Riemannian manifold. As applications, some optimalityconditions are obtained for optimization problems with constraints onRiemannian manifolds.

On the supporting quasi-hyperplane and separation theorem of geodesic convex sets with applications on Riemannian manifolds