\(C\)-nearest points and the drop property (Q1915564)

For a closed convex set \(C\) with non-empty interior, we define the \(C\)-nearest distance from \(x\) to a closed set \(F\). We show that, if there exists in the Banach space \(X\) a closed convex set with non-empty interior satisfying the drop property, then for all closed subsets \(F\) of \(X\), there exists a dense \(G_\delta\) subset \(\Gamma\) of \(X\backslash \{x: \rho(F, x)= 0\}\) such that every \(x\in \Gamma\) has a \(C\)-nearest point in \(F\). We also prove that every smooth (unbounded) convex set with the drop property has the smooth drop property.