An algorithm that works at least for dimensions $2$ and $3$ is:
- calculate a spanning tree of the $n$ points
- calculate the bisector planes of the spanning tree's edges
- the sought poiint is in the intersection of the affine subspace defined by the intersection of the bisector planes and has least distance to any of the given ones.
that algorithm can be generalized to arbitrary many points by taking the least squares solution to the intersection of the bisector planes with the subspace that is orthogonal to these intersections and contains th $n$.
For numerical reasons the maximum weight spanning tree seems preferable over the minimum weight spanning tree; it is also recommended to move the $n$ points center of gravity to the origin to prior to the calculations.