I want to compute the volume of region bounded by the inequalities : $x_2x_4x_6\geq x_1x_3x_5x_7, x_1x_4x_5\geq x_2x_3x_6x_7, x_3x_4x_7\geq x_1x_2x_5x_6, x_1x_2x_3\geq x_4x_5x_6x_7, x_2x_5x_7\geq x_1x_3x_4x_6, x_1x_6x_7\geq x_2x_3x_4x_5, x_3x_5x_6\geq x_1x_2x_4x_7, 0\leq x_i\leq 1.$
Is there any easy ways to compute this volume or a method to approximate the volume?
The easier version $x\geq yz,y\geq xz,z\geq xy,0<x,y,z\leq1$ can be solved by cutting the region defined by the inequality with the plane $z=a$ and then see what the plane section looks like.