Monge point

Remarks

For a tetrahedron, $n = 3$ and $m = 2g - o$, the reflection of $o$ in $g$. In that case $m$ is the common point of the six planes through the midpoint of an edge perpendicular to the opposite edge.

For a triangle, $n = 2$ and $m = 3g - 2o$, which is the orthocentre. Example: the triangle with vertices $(0, 0)$, $(2, 0)$ and $(0, 2)$ has $o = (1, 1)$ and $g = (\frac{2}{3}, \frac{2}{3})$. Then $m = (1, 1) + 3 \cdot (-\frac{1}{3}, -\frac{1}{3}) = (0, 0)$, the vertex of the right angle, which is indeed the orthocentre.

Non-example: for a segment, $n = 1$, the factor $\frac{n+1}{n-1}$ is undefined, so a $1$-simplex has no Monge point.

For a regular tetrahedron, $g = o$, so the Monge point is the centre.