2016 AMC 12B Problem 23

The set $|x|+|y|+|z|\le1$ is a regular octahedron with diagonals of length $2$ and volume $\tfrac43$

The second inequality is that octahedron translated up by $1;$ their overlap is a similar octahedron with half the linear size

The region $|x|+|y|+|z|\le1$ is a regular octahedron with vertices at $(\pm1,0,0),(0,\pm1,0),(0,0,\pm1),$ whose volume is $2\cdot\tfrac13\cdot(\sqrt2)^2\cdot1=\tfrac43.$ The second region is the same octahedron shifted up by $1.$ Their intersection is bounded by another regular octahedron with diagonals of length $1,$ half the linear dimensions of the first, so its volume is $\left(\tfrac12\right)^3\cdot\tfrac43=\tfrac16.$

Thus, the correct answer is A.