The 1-Center and 1-Highway problem

We study a variation of the 1-center problem in which, in addition to a single supply facility, we are allowed to locate a highway. This highway increases the transportation speed between any demand point and the facility. That is, given a set $S$ of points and $v>1$, we are interested in locating the facility point $f$ and the highway $h$ that minimize the expression $\max_{p\in S}d_{h}(p,f)$, where $d_h$ is the time needed to travel between $p$ and $f$. We consider two types of highways ({\em freeways} and {\em turnpikes}) and study the cases in which the highway's length is fixed by the user (or can be modified to further decrease the transportation time).