Inequalities characterizing coisotone cones in Euclidean spaces. (English) Zbl 1129.47042
A pointed closed convex cone of dimension \(d\) in Euclidean space \({\mathbb R}^d\) is called coisotone if it is generated by \(d\) vectors which have pairwise non-positive scalar product. The cone is latticial if any two vectors have a supremum (and hence an infimum) with respect to the order induced by the cone. The present paper characterizes the coisotone cones among the latticial cones by several equivalent inequalities involving the supremum and the infimum.
Reviewer: Rolf Schneider (Freiburg i. Br.)
MSC:
| 47H07 | Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces |
| 46A40 | Ordered topological linear spaces, vector lattices |
Citations:
Zbl 0760.52003References:
| [1] | 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. · Zbl 0387.46022 |
| [2] | J. B. Hiriart-Urruty, Unsolved problems: at what points is the projection mapping differentiable? Am. Math. Monthly 89(7) (1982), 456–458. · Zbl 0505.26004 |
| [3] | R. B. Holmes, Smoothness of certain metric projections on Hilbert space, Trans. Am. Math. Soc. 184 (1973), 87–100. · Zbl 0242.46050 |
| [4] | G. Isac, A. B. Németh, Monotonicity of metric projections onto positive cones of ordered Euclidean spaces, Arch. Math. 46(6) (1986), 568–576. · Zbl 0574.41037 |
| [5] | G. Isac, A. B. Németh, Every generating isotone projection cone is latticial and correct, J. Math. Anal. Appl. 147(1) (1990), 53–62. · Zbl 0725.46002 |
| [6] | 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. · Zbl 0719.46011 |
| [7] | G. Isac, A. B. Németh, Projection methods, isotone projection cones, and the complementarity problem, J. Math. Anal. Appl. 153(1) (1990), 258–275. · Zbl 0711.47030 |
| [8] | G. Isac, A. B. Németh, Isotone projection cones in eucliden spaces, Ann. Sci. Math Québec 16(1) (1992), 35–52. · Zbl 0760.52003 |
| [9] | G. Isac, L.-E. Persson, Inequalities Related to isotonicity of projection and antiprojection operators, Math. Inequalities Appl. 1(1) (1998), 85–97. · Zbl 0904.46010 |
| [10] | 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. · Zbl 1030.46020 |
| [11] | R. R. Phelps, Convex sets and nearest points, Proc. Am. Math. Soc. 8 (1957), 790–797. · Zbl 0078.35701 |
| [12] | R. R. Phelps, Convex sets and nearest points. II, Proc. Am. Math. Soc. 9 (1958), 867–873. · Zbl 0109.14901 |
| [13] | R. R. Phelps, Metric projection and the projection method in Banach spaces, SIAM J. Control Optim. 23(6) (1985), 973–977. · Zbl 0579.90099 |
| [14] | R. R. Phelps, The gradient projection method using Curry’s steplength, SIAM J. Control Optim. 24(4) (1986), 692–699. · Zbl 0603.90117 |
| [15] | 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. · JFM 65.0498.02 |
| [16] | E.H. Zarantonello, (1971) Projection on convex sets in Hilbert spaces and spectral theory, Ed., Acad. Press, New York. · Zbl 0281.47043 |
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