The convex dimension of a graph

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