Measure of overall misfit between two polyhedra

Question

Imagine I have two arbitrary polyhedrons with the same volume. How could one reasonably measure the misfit between them. E.g. how could one determine the minimum possible volume that they could not share simultaneously?

Answer 1

Assume that we have polyhedrons $A$ and $B$ and the volume of each polyhedron is $1$. For instance, by misfit between them we can use the function $m(A,B)=\inf\mu(A\Delta i(B))$, where $\Delta$ is the symmetric difference (that is $X\Delta Y=(X\cup Y)\setminus (X\cap Y)$), $\mu$ is the volume, and the infimum is taken over all isometric copies $i(B)$ of the set $B$.