The Metric Projection

Abstract

In this chapter we shall study the various properties of the metric projection onto a convex Chebyshev set K. It is always true that Pk is nonexpansive and, if K is a subspace, even linear. There are a substantial number of useful properties that Pk possesses when K is a subspace or a convex cone. For example, every inner product space is the direct sum of any Chebyshev subspace and its orthogonal complement. More generally, a useful duality relation holds between the metric projections onto a Chebyshev convex cone and onto its dual cone. The practical advantage of such a relationship is that determining best approximations from a convex cone is equivalent to determining them from the dual cone. The latter problem is often more tractable than the former. Finally, we record a reduction principle that allows us to replace one approximation problem by another one that is often simpler.