Densely connected ambiguous loci of the metric projection in Hilbert spaces (Q1925020)

Let \(E\) denote a real separable Hilbert space of dimension greater than one, \(K\) (resp. \({\mathcal C})\) the space of all non-empty compact (resp. compact convex) subsets of \(E\) endowed with the Hausdorff metric. For \(X,Y\in K\) and \(y\in E\), the sets \(\pi_x(Y)\) and \(\pi_x(y)\) defined by \[ \begin{aligned} \pi_x(Y) &= \bigl\{x\in X: \text{ there is } y\in Y\text{ such that }|x-y|= \gamma(X,Y)\bigr\},\\ \pi_x(y) &= \bigl\{x\in X: |x-y|= \gamma(x,y) \bigr\},\end{aligned} \] where \(\gamma(X,Y)= \inf\{|x-y|: x\in X,y\in Y\}\) and \(\gamma(x,y) =\inf\{|x-y|: x\in X\}\), are called, respectively, metric projection of \(Y\) onto \(X\), and metric projection of \(y\) onto \(x\). For \(X\in K\), the sets \({\mathcal C}_a(X)\) and \({\mathcal C}_n(X)\) given by \({\mathcal C}_a (X)= \{Y\in{\mathcal C}: \text{diam} \pi_X(Y) \cdot\text{diam} \pi_Y(X)>0\}\). \({\mathcal C}_n(X) =\{Y\in {\mathcal C}: \text{diam} \pi_X(Y)+\text{diam} \pi_Y(X)=0\}\) are called, respectively, ambiguous locus of \(x\) in \({\mathcal C}\), and uniqueness locus of \(X\) in \({\mathcal C}\). A subset \(X\) of a metric space \(M\) is called densely connected in \(M\) if for every \(x\in M\) and \(r>0\) the set \(X \cap \{y\in M: d(x,y)<r\}\) contains an arc. A subset \(x\) of a complete metric space \(M\) is called residual in \(M\) if \(M\backslash x\) is of Baire first category in \(M\). The paper mainly deals with ambiguous and uniqueness loci. Main results: (i) (Theorem 3.1): There exists a residual subset \(K_0\) of \(K\) such that, for every \(X\in K_0\), the ambiguous locus \({\mathcal C}_a(X)\) is densely connected in \({\mathcal C}\). (ii) (Theorem 4.3): Let \(A\in K\) and suppose \(\dim E= \infty\). Then the uniqueness locus \({\mathcal C}_n(A)\) is a residual subset of \({\mathcal C}\).