Given a graph G and a set of vertices W subset of V(G), the Steiner interval of W is the set of vertices that lie on some Steiner tree with respect to W. A set U subset of V(G) is called g(3)-convex in G. if the Steiner interval with respect to any three vertices from U lies entirely in U. Henning et al. (2009)[5] proved that if every j-ball for all j >= 1 is g(3)-convex in a graph G, then G has no induced house nor twin C(4), and every cycle in G of length at least six is well-bridged. In this paper we show that the converse of this theorem is true, thus characterizing the graphs in which all balls are g(3)-convex. (C) 2011 Elsevier Ltd. All rights reserved.