The star-forming regions in Chamaeleon (Cha) are among the nearest (distance ~165 pc) and youngest (age ~2 Myrs) conglomerates of recently formed stars and among the ideal targets for studies of star formation. We search for new, hitherto unknown binary or multiple-star components and investigate their membership in Cha and their gravitationally bound nature. We used the NACO instrument at the VLT UT 4/YEPUN of the Paranal Observatory, at 2 or 3 different epochs, in order to obtain relative and absolute astrometric measurements, as well as differential photometry in the J, H, and Ks band. On the basis of known proper motions and these observations, we analysed the astrometric results in proper motion diagrams to eliminate possible (non-moving) background stars and establish co-moving binaries and multiples. DI Cha turns out to be a quadruple system with a hierachical structure, consisting of two binaries: a G2/M6 pair and a co-moving pair of two M5.5 dwarfs. For both pairs we detected orbital motion (P~130 and ~65 years), although in opposite directions. Sz 22 is a binary whose main component is embedded in a circumstellar disc or reflection nebula, accompanied by a co-moving M4.5 dwarf. CHXR 32 is a triple system, consisting of a single G5 star, weakened by an edge-on disc and a co-moving pair of M1/M3.5 dwarfs whose components show significant variations in their angular separation. Finally, Cha Halpha 5 is a binary consisting of two unresolved M6.5 dwarfs whose strong variations in position angle at its projected separation of only 8 AU imply an orbital period of ~46 years. DI Cha D and Cha Halpha 5 A&B are right at the stellar mass limit and could possibly be brown dwarfs. In spite of various previously published studies of the star-forming regions in Cha we found four hitherto unknown components in young low-mass binaries and multiple systems. (abridged)
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