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Algorithm to calculate moments of uniform distribution on convex polyhedra

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 ...