There is system of linear inequalities $$ Ax \leq K, $$ $$ x\geq a, x\leq b. $$ $A$ is $(n\times m)$-matrix, where $n\approx 100$ and $m\approx 10000$, $rank(A)=n$.
Suppose that on set of solutions for inequalities uniform probability measure is defined.
Is there efficient algorithm to calculate
1) Volume of polyhedron, i.e. normalization factor for distribution.
2) Mean of distribution.
3) Second moment of distribution.
In other words these three integrals.
$$ \int_{Ax \leq K, x \geq a, x \geq b} dx $$ $$ \int_{Ax \leq K, x \geq a, x \geq b} x dx $$ $$ \int_{Ax \leq K, x \geq a, x \geq b} x^2 dx $$
