  We introduce the \emph{visibility center} of a set of points inside a polygon
-- a point $c_V$ such that the maximum geodesic distance from $c_V$ to see any
point in the set is minimized. For a simple polygon of $n$ vertices and a set
of $m$ points inside it, we give an $O((n+m) \log {(n+m)})$ time algorithm to
find the visibility center. We find the visibility center of \emph{all} points
in a simple polygon in $O(n \log n)$ time.
  Our algorithm reduces the visibility center problem to the problem of finding
the geodesic center of a set of half-polygons inside a polygon, which is of
independent interest. We give an $O((n+k) \log (n+k))$ time algorithm for this
problem, where $k$ is the number of half-polygons.
