I am looking for a reference for the volume of the simplex in N dimensions between the points N on every axis.
E.g in 2 dimensions the line between the points (0,2) und (2,0). the length is $$\sqrt{2} \cdot 2$$
In N dimensions it is: $$\sqrt{N} \frac{N^{N-1}}{(N-1)!}$$
Has anybody a reference for this formula?
Thanks!
Add the origin as another vertex to obtain a $N$-dimensional simplex, Its volume is (up to a constant) the product of base "area" and height. On the one hand, the height is $N$ and the base a scaled standard $(N-1)$-simplex, which has $(N-1)$-dimensional volume $\frac 1{(N-1)!}\cdot N^{N-1}$. On the other hand, this is the desired $(N-1)$-dimensional volume $V$ times the distance of its hyperplane from the origin (which is readily found to be $\sqrt N$).