Abstract
We introduce the concept of a φ-convex subset of a Hadamard manifold. Then we prove that for a φ-convex subset S of a Hadamard manifold M there exists an open set U containing S such that the metric projection is a single valued locally Lipschitz mapping on U.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Azagra, D., Ferrera, J., López-Mesas, F.: Nonsmooth analysis and Hamilton-Jacobi equation on Riemannian manifolds. J. Funct. Anal. 220, 304–361 (2005)
Azagra, D., Ferrera, J.: Proximal calculus on Riemannian manifolds. Mediterr. J. Math. 2, 437–450 (2005)
Canary, R.D., Epstein, D.B.A., Marden, A.: Fundamentals of Hyperbolic Geometry: Selected Expositions. Cambridge University Press, Cambridge (2006)
Canino, A.: On p-convex sets and geodesics. J. Differ. Equ. 75, 118–157 (1988)
Canino, A.: Existence of a closed geodesic on p-convex sets. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5, 501–518 (1988)
Colombo, G., Goncharov, V.: Variational inequalities and regularity properties of closed sets in Hilbert spaces. J. Convex Anal. 8, 197–222 (2001)
Colombo, G., Marigonda, A.: Differentiability properties for a class of non-convex functions. Calc. Var. Partial Differ. Equ. 25, 1–31 (2005)
Degiovanni, M., Marino, A., Tosques, M.: General properties of (p,q)-convex functions and (p,q)-monotone operators. Ric. Mat. 32, 285–319 (1983)
Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)
Da Cruz Neto, J.X., Ferreira, O.P., Lucambio Pérez, L.R.: Contributions to the study of monotone vector fields. Acta Math. Hung. 94, 307–320 (2002)
Greene, R.E., Shiohama, K.: Convex functions on complete noncompact manifolds: topological structure. Invent. Math. 63, 129–157 (1981)
Grognet, S.: Théorème de Motzkin en courbure négative. Geom. Dedic. 79, 219–227 (2000)
Hosseini, S., Pouryayevali, M.R.: Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds. Nonlinear Anal. 74, 3884–3895 (2011)
Lang, S.: Fundamentals of Differential Geometry. Graduate Text in Mathematics, vol. 191. Springer, New York (1999)
Ledyaev, Yu.S., Zhu, Q.J.: Nonsmooth analysis on smooth manifolds. Trans. Am. Math. Soc. 359, 3687–3732 (2007)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory, Grundlehren Series. Fundamental Principles of Mathematical Sciences, vol. 331. Springer, Berlin (2006)
Walter, R.: On the metric projection onto convex sets in Riemannian spaces. Arch. Math. (Basel) 25, 91–98 (1974)
Shapiro, A.: Perturbation analysis of optimization problems in Banach spaces. Numer. Funct. Anal. Optim. 13, 97–116 (1992)
Author information
Authors and Affiliations
Corresponding author
Additional information
The third author was partially supported by the Center of Excellence for Mathematics, University of Shahrekord, Iran.
Rights and permissions
About this article
Cite this article
Barani, A., Hosseini, S. & Pouryayevali, M.R. On the metric projection onto φ-convex subsets of Hadamard manifolds. Rev Mat Complut 26, 815–826 (2013). https://doi.org/10.1007/s13163-011-0085-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13163-011-0085-4