Nearest points and delta convex functions in Banach spaces

Given a closed set $C$ in a Banach space $(X,\mathrm{\parallel }\cdot \mathrm{\parallel })$, a point $x\in X$ is said to have a nearest point in $C$ if there exists $z\in C$ such that $d_C(x)=\mathrm{\parallel }x-z\mathrm{\parallel }$, where $d_C$ is the distance of $x$ from $C$. We shortly survey the problem of studying how large is the set of points in $X$ which have nearest points in $C$. We then discuss the topic of delta-convex functions and how it is related to finding nearest points.