On a generalized best approximation problem
Let \(C\) be a closed bounded convex subset of a Banach space \(E\) which has the origin of \(E\) as an interior point and let \(P_C\) denote the Minkowski functional with respect to \(C.\) For a nonempty closed set \(X\subset E\) and a point \(u\in E\) a minimization problem which consists in proving the existence of a point \(\widetilde x\in X\) such that \(P_C({\widetilde x}-u)= \inf\{P_C(x-u)\mid x\in X\}\) is studied . If \(\widetilde x\) is unique and every minimizing sequence converges to \(\widetilde x\), the minimization problem for \(u\in E, X\subset E\) is called well posed. It is proved that if \(E_0(X)\) is the set of all \(u\in E\) for which the minimization problem is well posed and the modulus of convexity with respect to \(P_C\) is strictly positive, then \(E_0(X)\) is a dense \(G_\delta\) subset of \(E\). It should be noted that for most bounded subsets \(X\subset E,\) the set \(E\backslash E_0(X)\) is dense in \(E.\)
scientific article; zbMATH DE number 1214032
scientific article