Given a graph G and a set S⊆V(G), the Δ-interval of S, SΔ, is the set formed by the vertices of S and every w∈V(G) forming a triangle with two vertices of S. If SΔ=S, then S is Δ-convex of G; if SΔ=V(G), then S is a Δ-interval set of G. The Δ-interval number of G is the minimum cardinality of a Δ-interval set and the Δ-convexity number of G is the maximum cardinality of a proper Δ-convex subset of V(G). In this work, we show that the problem of computing the Δ-convexity number is W1-hard and NP-hard to approximate within a factor O(n1−ɛ) for any constant ɛ>0 even for graphs with diameter 2 and that the problem of computing the Δ-interval number is NP-complete for general graphs. For the positive side, we present characterizations that lead to polynomial-time algorithms for computing the Δ-convexity number of chordal graphs and for computing the Δ-interval number of block graphs. We also present results on the Δ-hull, Δ-interval and Δ-convexity numbers concerning the three standard graph products, namely, the Cartesian, the strong and the lexicographic products, in function of these and well-studied parameters of the operands.