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Journal of Convex Analysis 18 (2011), No. 3, 707--720
Copyright Heldermann Verlag 2011

About the Existence of an Isotone Retraction onto a Convex Cone

Sandor Z. Németh
School of Mathematics, University of Birmingham, The Watson Building, Edgbaston - Birmingham B15 2TT, England

Alexandru B. Németh
Faculty of Mathematics and Computer Science, Babes-Bolyai University, Mihail Kogalniceanu Nr. 1, 400084 Cluj-Napoca, Romania

The existence of continuous isotone retractions onto pointed closed convex cones in Hilbert spaces is studied. The cones admitting such mappings are called isotone retraction cones. In finite dimension, generating, isotone retraction cones are polyhedral. For a closed, pointed, generating cone in a Hilbert space the isotonicity of a retraction and its complement implies that the cone is latticial and the retraction is well defined by the latticial structure. The notion of sharp mapping is introduced. If the cone is generating and normal, it is proved that its latticiality is equivalent to the existence of an isotone retraction onto it, whose complement is sharp. The subdual and autodual latticial cones are also characterized by isotonicity. This is done by attempting to extend Moreau's theorem to retractions.

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