  Let $L$ be a multidimensional L\'evy process under $P$ in its own filtration.
The $f^q$-minimal martingale measure $Q_q$ is defined as that equivalent local
martingale measure for $\mathcal {E}(L)$ which minimizes the $f^q$-divergence
$E[(dQ/dP)^q]$ for fixed $q\in(-\infty,0)\cup(1,\infty)$. We give necessary and
sufficient conditions for the existence of $Q_q$ and an explicit formula for
its density. For $q=2$, we relate the sufficient conditions to the structure
condition and discuss when the former are also necessary. Moreover, we show
that $Q_q$ converges for $q\searrow1$ in entropy to the minimal entropy
martingale measure.
