The Metric Projection

  • Frank Deutsch
Part of the CMS Books in Mathematics / Ouvrages de mathématiques de la SMC book series (CMSBM)


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.


Hilbert Space Linear Operator Product Space Convex Cone Nonempty Subset 


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Copyright information

© Springer-Verlag New York, Inc. 2001

Authors and Affiliations

  • Frank Deutsch
    • 1
  1. 1.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

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