$C$-nearest points and the drop property

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 subset $F$ of $X$, there exists a dense $G_\delta$ subset $\Gamma$ of $X\setminus \{x; \rho(F, x) = 0\}$ 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.