I have given two Polyhedrons $P$ and $Q$. What is the volume of their product $P \times Q = \{(p,q) \; | \; p \in P, q \in Q \}$. If $P$ and $Q$ are cubes, then it would be $vol(P \times Q) = vol(P) \cdot vol(Q)$. Is this also true for general $P$ and $Q$?
What I really want to calculate is the volume of something like $\{ (x_1,x_2,x_3,x_4) \in [a,b]^2 \times [c,d]^2 \; | \; x_1 \leq x_2, x_3 \leq x_4\}$ which I can write as $A \times B$ with $A = \{ (x,y) \in [a,b]^2 \; | \; x \leq y\}$ and $B = \{ (x,y) \in [c,d]^2 \; | \; x \leq y\}$. The volume of $A$ and $B$ are easy to compute since those are just triangles: $vol(A) = 1/2 (b-a)^2$ and $vol(B) = 1/2 (d-c)^2$.
Is the volume of their product $vol(A \times B) = vol(A) \cdot vol(B)$? And would it be the same for more dimensions and polytopes?
Thanks.