For two vertices u and v in a connected graph G, the set I(u, v) consists of all those vertices lying on a u − v geodesic in G. For a set S of vertices of G, the union of all sets I(u, v) for u, v ∈ S is denoted by I(S). A set S is convex if I(S) = S. The convexity number con(G) is the maximum cardinality of a proper convex set in G. A convex set S is maximum if |S| = con(G). The cardinality of a maximum convex set in a graph G is the convexity number of G. For a nontrivial connected graph H, a connected graph G is an H-convex graph if G contains a maximum convex set S whose induced subgraph is S = H. It is shown that for every positive integer k, there exist k pairwise nonisomorphic graphs H1, H2,...,Hk of the same order and a graph G that is Hi-convex for all i (1 ≤ i ≤ k). Also, for every connected graph H of order k ≥ 3 with convexity number 2, it is shown that there exists an H-convex graph of order n for all n ≥ k + 1. More generally, it is shown that for every nontrivial connected graph H, there exists a positive integer N and an H-convex graph of order n for every integer n ≥ N.