  A triple system is cancellative if it does not contain three distinct sets
$A,B,C$ such that the symmetric difference of $A$ and $B$ is contained in $C$.
We show that every cancellative triple system $\mathcal{H}$ that satisfies
certain inequality between the sizes of $\mathcal{H}$ and its shadow must be
structurally close to the balanced blowup of some Steiner triple system. Our
result contains a stability theorem for cancellative triple systems due to
Keevash and Mubayi as a special case. It also implies that the boundary of the
feasible region of cancellative triple systems has infinitely many local
maxima, thus giving the first example showing this phenomenon.
