In this paper we show that the set of right K-Lipschitz mappings from an asymmetric normed linear space (X,p) to another asymmetric normed linear space (Y,q), which vanish at a fixed point x 0 ∈ X can be endowed with the structure of an asymmetric normed cone. This provides an appropriate setting to characterize both the points of best approximation in asymmetric normed linear spaces. We also show that this space is bicomplete quasi-metric space.