How might one formulate integration over an ordered probability simplex?

Question

Given a probability simplex ($x_i \geq 0$, $i=1,\ldots,n$) \begin{equation} \sum_{i=1}^n x_i=1, \end{equation} I want to attempt integrations of various ("operator-monotone"-based https://pdfs.semanticscholar.org/d393/21f142432eddd2af0d3bd07235a63aca2019.pdf) functions over that subsection for which \begin{equation} x_i \geq x_{i-1}. \end{equation}

Further, I'm particularly interested in such integrations ($n=4$) with the additional ("absolute separability") constraint (eq. (3) in https://arxiv.org/pdf/quant-ph/0502170.pdf) \begin{equation} x_1 \leq x_3+2 \sqrt{x_2 x_4}. \end{equation}

What are my options for setting up the integrations (coordinate transformations might be of interest)? The particular integrations I have in mind are seemingly very challenging, and I may (probably) have to resort to numerical methods.

Answer 1

Using the GenericCylindricalDecomposition command of Mathematica, we can express the integral of 1 over the 3-dimensional probability simplex as the sum of three integrations (for ease of presentation, I replace the subscripted variables $x_i$ by $xi$, $i=1,2,3$)

Integrate[1, {x3, 0, 1/4}, {x2, x3, 1/2 (1 - 2 x3)}, {x1, 1 - x2 - 2 x3, 1 - x2 - x3}] + Integrate[1, {x3, 0, 1/4}, {x2, 1/2 (1 - 2 x3), (1 - x3)/2}, {x1, x2, 1 - x2 - x3}] + Integrate[1, {x3, 1/4, 1/3}, {x2, x3, 1/2 (1 - x3)}, {x1, x2, 1 - x2 - x3}]

The results of the three integrations are $\left\{\frac{1}{192},\frac{1}{768},\frac{1}{2304}\right\}$, respectively, summing to $\frac{1}{144}$, which is one-twenty-fourth ($\frac{1}{4!}$)of the integration ($\frac{1}{6}$) over the unordered simplex.

It remains now for my investigations to incorporate the absolute separability constraint, \begin{equation} x_1 \leq x_3+2 \sqrt{x_2 x_4}, \end{equation} into the integrations.