TITLE: closest equidistant point to N points in M dimensions
QUESTION [5 upvotes]: 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.

REPLY [4 votes]: 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 $N-(M+1)$.