The convex dimension of a graph

The convex dimension   of a graph G=(V,E)G=(V,E) is the smallest dimension dd for which GG admits an injective map f:V⟶Rdf:V⟶Rd of its vertices into dd-space, such that the barycenters of the images of the edges of GG are in convex position. The strong convex dimension   of GG is the smallest dd for which GG admits a map as above such that the images of the vertices of GG are also in convex position. In this paper we study the convex and strong convex dimensions of graphs.