On approximation properties of sets with convex complement
Let \(X\) be a Banach space and \(M\) a non-empty closed subset of \(X\). If \(x\in X\) then \(\phi(x)\) denotes the distance of \(x\) from \(M\) and \(P_M(x)\) denotes the set of points \(z\in M\) such that \(|x-z|= \phi(x)\). One denotes by \(\text{wAC}(M)\) the set of points \(x\in X\) of `weak approximate compactness' relative to \(M\) and by \({\mathcal F}_{CC}\) the family of non-empty closed subsets of \(X\) which have non-empty convex complements. If \(M\in {\mathcal F}_{CC}\) then \(-\phi\) is a convex function and if, further, at a point \(x\in X\backslash M\) the function \(\phi\) is Gâteaux differentiable with Gâteaux derivative \(\phi'(x)\in X^*\) then the hyperplane \(N=\{z\in X:\phi'(x)(z)= \phi'(x)(z)-\phi(x)\}\) is a subset of \(M\); \(\phi(x)\) is equal to the distance of \(x\) from \(N\) if and only if \(|\phi'(x)|=1\); and \(P_M(x)\) is non-empty if and only if \(|\phi'(x)|=1\) and \(\phi'(x)\) attains its bound. The paper explores this situation. A number of similar results are obtained of which the following is representative. If \(X\) is a Banach space then the following statements are equivalent: (1) \(X\) is reflexive, (2) for each \(M\in {\mathcal F}_{CC}\) the set \(E_M=\{x: P_M(x)\neq \emptyset\}\) is dense in \(X\), (3) for each \(M\in{\mathcal F}_{CC}\) the set \(X\backslash \text{wAC}(M)\) is of first category. It is further shown that if \(X\) is a Banach space and \(\eta\in [0,1)\) then \(\dim X=\infty\) if and only if there exist \(M\in {\mathcal F}_{CC}\) and a point \(x\in X\backslash M\) at which the Gâteaux derivative \(\phi'(x)\) exists and \(|\phi'(x)|=\eta\).
scientific article; zbMATH DE number 846455
scientific article