This may be in my urgent, I'm currently dealing with an integration problem over the support of high dimensional [0, 1] cubic. The integration function itself is not that complicated, you can assume that

F[p1_, p2_, p3_, p4_, p5_, p6_, p7_, p8_, p9_, p10_] := p1 * p2 * p3 * p4 * p5 * p6 * p7 * p8 * p9 * p10

But the region over cubic for integration is defined by the union of the following two region, i.e. $R_1 \cup R_2$:

$R_1: \prod_{i=1}^{s} p_{(i)} \leq c_1$

$R_2: \prod_{i=1}^{s} \frac{p_{(i)} - p_{(s+1)}}{1 - p_{(s+1)}} \leq c_2$

where $p_{(1)} \geq p_{(2)} \geq ...... \geq p_{(10)}$ is the ordered value of $(p_1, p_2, ... , p_{10})$, $0 < p_i < 1$ and $2 \leq s \leq 8$. $c_1, c_2$ are rather small value, for example, $c_1 = 0.0025, c_2 = 10^{-6}$.

I'm a primary learner in Mathematica, and could define some simple region only (like sphere, rectangle), but the region $R_1 \cup R_2$ is more complex than I expected, so I'm urgent to know how to define the region that ruled by relationship of ordering values.

Thanks very much!!

This may be in my urgent, I'm currently dealing with an integration problem over the support of high dimensional [0, 1] cubic. The integration function itself is not that complicated, you can assume that

F[p1_, p2_, p3_, p4_, p5_, p6_, p7_, p8_, p9_, p10_] := p1 + 2*p2 + 3*p3 + 4*p4 + 5*p5 + 6*p6 + 7*p7 + 8*p8 + 9*p9 + 10*p10

But the region over cubic for integration is defined by the union of the following two region, i.e. $R_1 \cup R_2$:

$R_1: \prod_{i=1}^{s} p_{(i)} \leq c_1$

$R_2: \prod_{i=1}^{s} \frac{p_{(i)} - p_{(s+1)}}{1 - p_{(s+1)}} \leq c_2$

where $p_{(1)} \geq p_{(2)} \geq ...... \geq p_{(10)}$ is the ordered value of $(p_1, p_2, ... , p_{10})$, $0 < p_i < 1$ and $2 \leq s \leq 8$. $c_1, c_2$ are rather small value, for example, $c_1 = 0.0025, c_2 = 10^{-6}$.

I'm a primary learner in Mathematica, and could define some simple region only (like sphere, rectangle), but the region $R_1 \cup R_2$ is more complex than I expected, so I'm urgent to know how to define the region that ruled by relationship of ordering values.

Thanks very much!!