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Is there a way to find the volume of the "feasible region" of a standard simplex satisfying simple range constraints?

$x_1+x_2+...+x_n = 1$
$a_1 \le x_1 \le b_1$
$a_2 \le x_2 \le b_2$
$...$
$a_n \le x_n \le b_n$

where $a_i, b_i \in ]0, 1[$, $a_i \le b_i$ are the ranges for each $x_i$.

Of course some of the inequality constraints may be redundant, and the volume can be zero (if there is no solution for the given constraints). Thanks a lot!

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    $\begingroup$ As far as I know, this is difficult even with the help of a computer. If you want a numerical approximation, there exist good probabilistic algorithm, among which "hit and run" which should lead you to relevant publications. However I am far from being a specialist on the domain, maybe other users will have real answers. $\endgroup$ – Benoît Kloeckner Sep 2 '15 at 8:10
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Well, there is Brion's formula. (see also Barvinok's book)

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