This does not answer your (interesting) question, but it may be worth mentioning that
this variant (which does not use geodesic distance) has been studied:
Thomas Shermer, "Hiding people in polygons,"
Volume 42, Numbers 2-3 (1989), 109-131 (Springer link):
A hidden set is a set of points such that no two points in the set are visible to each other. A hidden guard set is a hidden set which is also a guard set. In this paper we consider the problem for finding hidden sets and hidden guard sets in and around polygons. In particular, we establish bounds on the maximum size of hidden sets, and show that the problem of finding a maximum hidden set is NP-hard.
To more directly address your question, perhaps this is the most natural interpretation for $m=1$:
Pollack, Sharir, Rote, "Computing the Geodesic Center of a Simple Polygon,"
Discrete Comput. Geom. 4:611-626 (1989):
The geodesic center of a simple polygon is a point inside the polygon which minimizes the maximum internal distance to any point in the polygon. We present an algorithm which calculates the geodesic center of a simple polygon with $n$ vertices in time $O(n \log n)$.