Bound on volume of Minkowski Difference

$ A-B:=\{c:B+c\subseteq A\}.$

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$ further reduces $ vol(A-B)$ by set theoretic consideration.) Is there anything tighter?