Let X be a normed vector space, let C be a subset of X and let Pc(z) denote the set of projections of a point zεX onto CAssume that C is proximinal, that is, for all zεX Pc(z) is nonempty. C will be called unified if zεX[image omitted] , and μεC imply [image omitted] . In the context of estimation, μ is an unknown parameter assumed apriori to be in C. and z is an unconstrained data-based estimate of μ. If C is unified, then [image omitted] is closer to both μεC (assumption) and z (data) than μ and z are to themselves. If X is an inner product space, then C is unified if and only if it is proximinal and convex. This is not the case, how-ever, if the norm does not admit an inner product. In this paper we characterize those spaces in which proximinal and convex imply unified, and partially characterize spaces in which unified implies convex. In particular, if X is complete and has three or more dimensions, then proximinal and convex imply unified only if X has an inner product. An example involving M-estimation is discussed.
Tijdschrift beschrijving