  As the reversed version of usual symmetric norms, we introduce the notion of
symmetric anti-norms $\|\cdot\|_!$ defined on the positive operators affiliated
with a finite von Neumann algebra with a finite normal trace. Related to
symmetric anti-norms, we develop majorization theory and superadditivity
inequalities of the form $\|\psi(A+B)\|_!\ge\|\psi(A)\|_!+\|\psi(B)\|_!$ for a
wide class of functions $\psi$.
