In this paper, L-convex β∗-remotehood system is introduced and some characterizations of both L-convexity and L-convex remotehood system are obtained. Further, β∗-remote mapping is presented and some of its properties are investigated. Based on this, L-convex quasi-uniformity and L-convex quasi- uniformity preserving mapping are introduced. It is proved that L-convex quasi-uniform space and L- convex space are mutually induced. In addition, the category of L-convex spaces and the category of L-convex β∗-remotehood spaces can be embedded into the category of L-convex quasi-uniform spaces.