Intersection of $(n-1)$-unit simplex and $n$-orthotope

Question

Let $\Delta^{n-1}=\{\mathbf{x}\in \mathbb{R}^{n}_{+}|\sum_{i=1}^n x_i=1 \}$ be an $(n-1)$-unit simplex and $T_n=\{(x_1,\dots,x_n)\in \prod_{i=1}^n[0,a_i]|0<a_i\le 1\}$ be an $n$-orthotope. I want to get the volume of their intersection i.e. $V(\Delta^{n-1}\cap T_n)$ for any $n$.

Any hints or relevant references are appreciated.

Answer 1

Only a partial answer.

Consider the regular hypercube $x4o3o3o3...3o$ instead of the general orthotope (hyper-brique). Then the sectioning plane runs through orthogonally to the diagonal. The section itself then can be described (having scaled down everything by sqrt(2) throughout in the followings):

• $o3o3o3...3o$ at the first instance
• $y3o3o3...3o$ up to the first vertex layer, with $0<y<1$
• $x3o3o3...3o$ at the level of that first vertex layer
• $(x-y)3y3o3...3o$ between the first and second vertex layer, with $0<y<1$
• $o3x3o3...3o$ at the level of the second vertex layer
• $o3(x-y)3y3...3o$ between the second and third vertex layer, with $0<y<1$
• etc.
• $o3o3o3...3(x-y)$ up to the last (opposite) vertex, with $0<y<1$
• $o3o3o3...3o$ at the last (opposite) vertex again

Throughout I use $x$ as a typewriter friendly replacement of a ringed node of the Coxeter-Dynkin diagram, $o$ the according unringed node. Moreover I assume that the according edge size of that $x$ is 1. $y$ will be some other edge type then with a size as specified. And $(x-y)$ would specify an accordingly decreasing edge while $y$ increases, for sure.

Thus the remainder then would be simply to calculate the according volumes of the various truncates or rectates for varying edge sizes.

--- rk