  If D is a big divisor and E is an effective divisor, then
vol(D-E)<=vol(D)<=vol(D+E). We discuss when each inequality is an equality.
Surprisingly, the answer is that the asymptotic equality vol(D-E)=vol(D) is
equivalent to the equality of the Hilbert functions of D-E and D. The analogous
result holds when vol(D)=vol(D+E).
