  Based on heuristic arguments we conjecture that an intimate relation exists
between the eigenfunction multifractal dimensions $D_q$ of the eigenstates of
critical random matrix ensembles $D_{q'} \approx qD_q[q'+(q-q')D_q]^{-1}$,
$1\le q \le 2$. We verify this relation by extensive numerical calculations. We
also demonstrate that the level compressibility $\chi$ describing level
correlations can be related to $D_q$ in a unified way as
$D_q=(1-\chi)[1+(q-1)\chi]^{-1}$, thus generalizing existing relations with
relevance to the disorder driven Anderson--transition.
