Viviani's theorem claims that given a point inside an equilateral triangle, the sum of the distances from the point to the sides of the triangle is constant an equal to the height of the triangle. Also, it can be proven that the only 2D shape that satisfies the property that the sum the three distances is constant is an equilateral triangle.
Generalisations to 3D have been made, and for a regular tetrahedron, if an inner point is chosen, the sum of the distances from that point to the sides of the tetrahedron remains constant. In this case, nevertheless, there are more 3D polyhedrons that satisfy this property, as long as the faces have equal area.
And here it is my question: does Viviani's theorem hold for n-simplex in dimensions higher than 3? My intuition says that this is the case, but I cannot find any proof of such claim on the web.
Many thanks in advance!
The "volume" (area, volume, hypervolume, etc) of an $n$-simplex in $\mathbb{R}^n$ is given by $$\tfrac1n\cdot(\text{distance from a vertex to opposite facet})\cdot|\text{facet}|$$ (where "$|\;\;|$" indicates "volume", and a "facet" is an ($n-1$)-dimensional side of the simplex). Joining an interior point $P$ to the vertices determines $n+1$ subsimplices such that $$|\text{simplex}|=\sum_k |\text{subsimplex $k$}|=\sum_k\tfrac1n\cdot(\text{distance from $P$ to facet $k$})\cdot|\text{facet $k$}|$$ (We can allow $P$ to be exterior if we use signed distances to the facets.)
If the facets have equal volume, then we can divide-through by that volume (and $1/n$) to get that the sum of the distances from $P$ to the facets is a constant (namely, the "height" from a vertex), thus generalizing Viviani.