A convexity in ${\mathbb{R}}^2$ with river metric

In this paper we consider the space ${\mathbb{R}}^2$ with the river metric $d^{\ast }$ and different types of convexity of this space. We define $W$-convex structure in $({\mathbb{R}}^2,d^{\ast })$ and we give the complete characterization of the convex sets in this space. We consider some measures of noncompactness and we give the moduli of noncompactness for considered measures on this space.