$A-B:=\{c:B+c\subseteq A\}$, in, say, Euclidean space.
I think that if $A-B$ is not the universe, then $vol(A-B)\leq vol(A)$ (If $B$ has one point then the inequality is immediate, adding more points to $B$ can only reduce $vol(A-B)$ by set theoretic consideration.) Is there anything tighter?