Projections onto convex sets on the sphere
In this paper some concepts of convex analysis are extended in an intrinsic way from the Euclidean space to the sphere. In particular, relations between convex sets in the sphere and pointed convex cones are presented. Several characterizations of the usual projection onto a Euclidean convex set are extended to the sphere and an extension of Moreau’s theorem for projection onto a pointed convex cone is exhibited.
KeywordsSphere Pointed convex cone Convex set in the sphere Projection onto a pointed convex cone
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- 3.do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice-Hall Inc., Englewood Cliffs, N.J. (Translated from the Portuguese) (1976)Google Scholar
- 4.do Carmo, M.P.: Riemannian Geometry. Mathematics: Theory and Applications. Birkhäuser Boston Inc., Boston, MA (Translated from the second Portuguese edition by Francis Flaherty) (1992)Google Scholar
- 9.Moreau J.-J.: Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement polaires. C. R. Acad. Sci. Paris 255, 238–240 (1962)Google Scholar
- 12.Sakai, T.: Riemannian Geometry, Vol. 149 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI (Translated from the 1992 Japanese original by the author) (1996)Google Scholar
- 13.Smith, S.T.: Optimization techniques on Riemannian manifolds. In: Bloch, A. (ed.) Hamiltonian and Gradient Flows, Algorithms and Control, vol. 3 of Fields Inst. Commun., pp. 113–136. Am. Math. Soc., Providence, RI (1994)Google Scholar
- 18.Xue G.L.: Algorithms for Constrained Approximation and Optimization. Stowe, VT (1993)Google Scholar