Projections onto convex cones in Hilbert space
A subset C of a real Hilbert space H is called a Chebyshev set if each \(x\in H\) has a unique point p(x) of best approximation in C. It is not difficult to see that if C is a closed convex cone, then p is (1) idempotent, (2) non-negative homogeneous, and (3) \(p(x+y)=p(x)+p(y)\) if and only if \(<p(x),y>=<p(x),p(y)>=<x,p(y)>.\) The authors investigate the converse question, namely whether conditions (1), (2), and (3) on a mapping n: \(H\to H\) automatically force \(K=n(H)\) to be a closed convex cone with projection n. Amont other things, they show that \(K=n(H)\) is a closed, convex cone and n is the projection mapping of H onto K if and only if the function n is continuous and satisfies the three conditions. The paper concludes with a discussion of the open question of whether the assumption of continuity in the above result can be omitted.
scientific article; zbMATH DE number 4202997
scientific article