A total of 28 young nearby stars (ages <= 60 Myr) have been observed in the K-s-band with the adaptive optics imager Naos-Conica of the Very Large Telescope at the Paranal Observatory in Chile. Among the targets are ten visual binaries and one triple system at distances between 10 and 130 pc, all previously known. During a first observing epoch a total of 20 faint stellar or sub-stellar companion-candidates were detected around seven of the targets. These fields, as well as most of the stellar binaries, were re-observed with the same instrument during a second epoch, about one year later. We present the astrometric observations of all binaries. Their analysis revealed that all stellar binaries are co-moving. In two cases (HD 119022AB and FG Aqr B/C) indications for significant orbital motions were found. However, all sub-stellar companion candidates turned out to be non-moving background objects except PZ Tel which is part of this project but whose results were published elsewhere. Detection limits were determined for all targets, and limiting masses were derived adopting three different age values; they turn out to be less than 10 Jupiter masses in most cases, well below the brown dwarf mass range. The fraction of stellar multiplicity and of the sub-stellar companion occurrence in the star forming regions in Chamaeleon are compared to the statistics of our search, and possible reasons for the observed differences are discussed. (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim