Modulus of strong proximinality and continuity of metric projection
For a real Banach space \(X\) and a closed subspace \(Y\) of \(X,\) the metric projection is the set-valued mapping \(P_Y:X\rightrightarrows Y\) given by \(P_Y(x)=\{y\in Y : \|x-y\|=d(x,Y)\},\;x\in X\). For \(t>0\) put \(P_Y(x,t)=\{y\in Y : \|x-y\|< d(x,Y)+t\}\). The subspace \(Y\) is called proximinal if \(P_Y(x)\neq \emptyset\) for every \(x\in X\) and strongly proximinal at \(x\in X\) if for every \(\varepsilon >0\) there exists \(t>0\) such that \(P_Y(x,t)\subset P_Y(x)+\varepsilon B_Y,\) where \(B_Y\) denotes the closed unit ball of \(Y\). The modulus of strong proximinality of a proximinal subspace \(Y\) of \(X\) is defined by \(\varepsilon(x,t)=\inf\{r>0 : P_Y(x,t)\subset P_Y(x)+r\;B_Y\},\;x\in X\setminus Y,\;t>0\). A subspace \(Y\) of \(X\) is strongly proximinal iff \(\lim_{t\searrow 0}\varepsilon(x,t)=0\) for every \(x\in X\setminus Y\). Some semi-continuity properties of \(\varepsilon(\cdot,t)\) hold provided that the metric projection \(P_Y\) has some appropriate semi-continuity properties. Also, if \(Y\) is proximinal and \(P_Y\) is Hausdorff continuous at \(x\), then \(\varepsilon(\cdot,t)\) is continuous at \(x\). If \(Y\) is strongly proximinal, then the converse is also true. A proximinal subspace \(Y\) of \(X\) has the \(1\frac{1}{2}\)-ball property iff \(\varepsilon(x,t)<1\) for every \(x\in X\setminus Y\) (Theorem 3.2). Also, if \(X\) is uniformly convex with modulus of uniform convexity satisfying \(\delta_X(t)\geq Ct^p,\) for some \(p,\;2\leq p<\infty\), then there exists \(K>0\) (depending only on \(X\)) such that \(\varepsilon(x,t)\leq Kt^{1/p}\) for every \(x\in S_X\) with \(d(x,Y)=1\) and sufficiently small \(t>0\). If \(Y\) is a strongly proximinal subspace of \(C(K)\) of finite codimension, then for every \(f\in C(K)\setminus Y\) there exists \(C>0\) and \(t_0>0\) such that \(\varepsilon(f,t)\leq Ct\) for every \(0<t<t_0\).
scientific article; zbMATH DE number 5905337
scientific article