Nowadays, Peer-to-Peer (P2P) live streaming applications have attracted great interest. Despite the fact that numerous systems have been proposed in the past, there are still problems concerning delay and quality requirements of live video distribution. In this paper, we consider a pull-based P2P live video streaming system where the video is disseminated over an overlay network. We propose a mesh-based overlay construction mechanism that enhances the received video quality while minimizing, as much as possible, the play-out delay. The principal feature provides each newcomer with a set of neighbors holding almost the same video segments and enough available transmission capacities to deal with its requests. A particular algorithm has been designed to estimate the peer's available capacities. The results of simulations show our mechanism efficiency in heterogeneous systems. Keywords-Peer to peer networks; live video streaming; overlay network; play-out delay.