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I posted this question a while ago, but since I didn't have much luck I though I'd reformulate it and try again.

Question: What is the general form of the equation that gives the hypervolume under the square of an $n$-simplex in an $n+1$ dimensional space? The equation should be in terms of the area of the projection of the simplex and the distance between the vertices of the simplex and its projection.

I think an example and some figures would help.

2-D Case

In the figure, we have a 1-simplex. The projection of the simplex is $S=(x_{i},x_{i+1})$. The distances between the vertices of the simplex and its projection are $e_{i}$, and $e_{i+1}$.

If the distance between the simplex and a point $p$ on its projection is $d(p)$ ( so that $d(x_{i})=e_{i}$), then the hypervolume of the square distance under the simplex (area in the 2d case) is$$ V=\int_{p \epsilon S} d(p)^2 dS , $$ If we shift everything so that $x_{i}$ is at the origin then$$ V=\frac{|S|}{3}[e_{i}^2 + e_{i+1}^2 + e_{i}e_{i+1}] $$

where $|S|$ is the hypervolume (length in the 2d case) of $S$

figure

3-D Case

Again, the projection of the simplex is $S=(x_{i,j},x_{i+1,j},x_{i,j+1})$ and the distance between the vertices of the simplex and its projection are $e_{i,j}, e_{i+1,j}$, and $e_{i,j+1}$ The hypervolume of the square distance between the simplex and its projection is$$ V=\int\int_{p \epsilon S} d(p)^2 dS , $$ where $d(p)$ is the vertical distance between the simplex and a point $p$ on its projection (so that $d(x_{i,j+1})=e_{i,j+1}$). Again, if we shift $x_{i,j}$ to the origin$$ V=\frac{|S|}{6}[e_{i,j}^2 + e_{i+1,j}^2 + e_{i,j+1}^2 + e_{i,j}e_{i+1,j}+ e_{i,j}e_{i,j+1} + e_{i+1,j}e_{i,j+1}] $$ Again $|S|$ is the hypervolume (area in the 3d case) of the simplex projection. figure

I got the last equation from this paper but it doesn't say how they derived it. So my question is how to generalize equation for $V$ to the $n$-D case?

Any suggestions on where to look are much appreciated.

Thanks

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