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!!