closest equidistant point to N points in M dimensions Is there a formula/algorithm/etc. to find the closest equidistant point (assuming it exists) to a set of points, allowing that the number of dimensions of the space is independent of the number of points? Any help would be greatly appreciated.
 A: Let $p$ be the closest equidistant point to your set of points, $\{p_1,\dots,p_N\}$. Then $p$ is in the affine subspace $X$ generated by the points: if not, the orthogonal projection of $p$ in $X$ is also at the same distance to all the points and closer to the points than $p$, which is a contradiction.
$p$ is the unique point of $X$ at the same distance from $\{p_1,\dots,p_N\}$: if there was other such point $p'$, then all the $p_i$ would be in the intersection of two spheres of centers $p$ and $p'$, which is impossible because the $p_i$ affinely generate $X$ and the intersection of two distinct spheres in $X$ is always contained in a hyperplane of $X$.
So you can compute $p$ as the intersection of the affine subspace $X$ generated by $p_1,\dots,p_N$ and the bisector hyperplanes $H_{p_1,p_2},H_{p_2,p_3},\dots,H_{p_{n-1},p_N}$. Moreover, if your set of points is not affinely independent, you can change it by a maximal affinely independent subset. All of this assuming the existence of $p$, of course.
A: If such a point exists, then it is the center of a sphere on which all $N$ points lie, so I think your question is equivalent to asking, do all $N$ points lie on some sphere?  Since $M+1$ points determine a sphere in $\mathbb{R}^M$, I think it would be enough to first use $M+1$ points to determine that sphere (for example, as described in a/the answer here: https://math.stackexchange.com/questions/2611326/how-many-points-define-a-sphere-of-unknown-radius), and then check whether the remaining $N-(M+1)$ points are located at the given distance from the given center.
In short: the formula/algorithm merely involves solving a linear system of $M+1$ equations, and then checking distances of the remaining points $N-(M+1)$.
A: 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 the $n$ points.
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.
