Characterization of latticial cones in Hilbert spaces by isotonicity and generalized infimum (Q624250)

Let \(H\) be a Hilbert space and \(K\) be a cone of it. The following are the main results of the paper. (i) Assume that \(K\) is latticial. Then \(\inf\{x,y\}=P_{(x-K)\cap (y-K)} (1/2)(x+y)\) \(\forall x,y\in H\) iff \(K\) is subdual. (ii) Assume that \(K\) is pointed, closed, convex, generating and normal. Then \(K\) is latticial iff there exists a continuous isotone retraction \(\rho:H\to K\) such that its complement \(I-\rho\) is sharp. The obtained facts may have potential applications to complementarity theory.