For a convex set in Euclidean space, there is an obvious notion of its center: namely, its center of mass, which by convexity lies in the set. For a nonconvex set there is just as obviously no nice notion of its center, even if we restrict consideration to a connected component. (As a tangential aside, recall the balancing bird toy.)
To narrow things a bit, let's consider a connected (but not necessarily topologically trivial) region in the plane with smooth boundary. One notion of central points of this region is furnished by taking erosions by disks of varying radius; another is furnished by considering the heat equation with initial data 1 on the boundary and 0 in the interior, then looking for points of minimal temperature (it is not quite clear to me that these points are fixed over time, but the alternative seems very unlikely).
I am curious to know (if and) where such notions have been studied, and ideally surveyed/catalogued. Higher dimensions would be nice to know about too.