Circumsphere

Remarks

The point $o$ exists and is unique because the conditions $\norm{o - v_i}^2 = \norm{o - v_0}^2$ for $i = 1, \ldots, n$ are $n$ affine equations in $S$. Their linear parts $v_i - v_0$ are linearly independent, by affine independence of the vertices.

Example: the triangle with vertices $(0, 0)$, $(2, 0)$ and $(0, 2)$ has circumcentre $(1, 1)$ and circumradius $\sqrt{2}$. Each vertex is at distance $\sqrt{1 + 1} = \sqrt{2}$ from $(1, 1)$. For $n = 2$ this is the circumcircle.

Non-example: for four points $(0, 0)$, $(2, 0)$, $(0, 2)$ and $(3, 3)$ in the plane, there is no circle through all of them. The only circle through the first three has centre $(1, 1)$, and $(3, 3)$ is at distance $2\sqrt{2}$ from it. The definition needs the vertices of a simplex, which are affinely independent.

Without the condition $o \in S$, the centre would not be unique when $n < N$: every point of the affine subspace through $o$ orthogonal to $S$ is equidistant from the vertices.