Abstract
The isotone projection cone, defined by G. Isac and A. B. Németh, is a closed pointed convex cone such that the order relation defined by the cone is preserved by the projection operator onto the cone. In this paper the coisotone cone will be defined as the polar of a generating isotone projection cone. Several equivalent inequality conditions for the coisotonicity of a cone in Euclidean spaces will be given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Haraux, How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities, J. Math. Soc. Japan 29(4) (1977), 615–631.
J. B. Hiriart-Urruty, Unsolved problems: at what points is the projection mapping differentiable? Am. Math. Monthly 89(7) (1982), 456–458.
R. B. Holmes, Smoothness of certain metric projections on Hilbert space, Trans. Am. Math. Soc. 184 (1973), 87–100.
G. Isac, A. B. Németh, Monotonicity of metric projections onto positive cones of ordered Euclidean spaces, Arch. Math. 46(6) (1986), 568–576.
G. Isac, A. B. Németh, Every generating isotone projection cone is latticial and correct, J. Math. Anal. Appl. 147(1) (1990), 53–62.
G. Isac, A. B. Németh, Isotone projection cones in Hilbert spaces and the complementarity problem, Boll. Un. Mat. Ital. B 7(4) (1990), 773–802.
G. Isac, A. B. Németh, Projection methods, isotone projection cones, and the complementarity problem, J. Math. Anal. Appl. 153(1) (1990), 258–275.
G. Isac, A. B. Németh, Isotone projection cones in eucliden spaces, Ann. Sci. Math Québec 16(1) (1992), 35–52.
G. Isac, L.-E. Persson, Inequalities Related to isotonicity of projection and antiprojection operators, Math. Inequalities Appl. 1(1) (1998), 85–97.
A. B. Németh, Characterization of a Hilbert vector lattice by the metric projection onto its positive cone, J. Approx. Theory 123(2) (2003), 295–299.
R. R. Phelps, Convex sets and nearest points, Proc. Am. Math. Soc. 8 (1957), 790–797.
R. R. Phelps, Convex sets and nearest points. II, Proc. Am. Math. Soc. 9 (1958), 867–873.
R. R. Phelps, Metric projection and the projection method in Banach spaces, SIAM J. Control Optim. 23(6) (1985), 973–977.
R. R. Phelps, The gradient projection method using Curry’s steplength, SIAM J. Control Optim. 24(4) (1986), 692–699.
A. Youdine, Solution de deux problèmes de la théorie de espaces semi-ordonnés, C. R. Acad. Sci. U.R.S.S. 27 (1939), 418–422.
E.H. Zarantonello, (1971) Projection on convex sets in Hilbert spaces and spectral theory, Ed., Acad. Press, New York.
Author information
Authors and Affiliations
Corresponding author
Additional information
Thanks are due to A. B. Németh who draw the author’s attention on the relation of latticial cones generated by vectors with pairwise non-accute angles with the theory of isotone cones.
Rights and permissions
About this article
Cite this article
Németh, S.Z. Inequalities Characterizing Coisotone Cones in Euclidean Spaces. Positivity 11, 469–475 (2007). https://doi.org/10.1007/s11117-007-2109-3
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11117-007-2109-3