On a Generalized Best Approximation Problem

Abstract

LetCbe a closed bounded convex subset of a Banach spaceEwhich has the origin ofEas an interior point and letpCdenote the Minkowski functional with respect toC. Given a closed setX⊂Eand a pointu∈Ewe consider a minimization problem minC(u, X) which consists in proving the existence of a pointx∈Xsuch thatpC(x−u)=λC(u, X), whereλC(u, X)=inf{pC(x−u)∣x∈X}. If such a point is unique and every sequence {xn}⊂Xsatisfying the condition limn→+∞pC(xn−u)=λC(u, X) converges to this point, the minimization problem min(u, X) is called well posed. Under the assumption that the modulus of convexity with respect topCis strictly positive, we prove that for every closed subsetXofE, the setEo(X) of allu∈Efor which the minimization problem minC(u, X) is well posed is a residual subset ofE. In fact we show more, namely that the setE\\Eo(X) isσ-porous inE. Moreover, we prove that for most closed bounded subsetsXofE, the setE\\Eo(X) is dense inE.