Let \(\cal{F}\) be a family of sets in ℝd (always d ≥ 2). A set M ⊂ ℝd is called \(\cal{F}\) -convex, if for any pair of distinct points x, y ∈ M, there is a set \(F \in \cal{F}\) such that x, y ∈ F and F ⊂ M. We obtain the Γ-convexity, when \(\cal{F}\) consists of Γ-paths. A Γ-path is the union of both shorter sides of an isosceles right triangle. In this paper we first characterize some Γ-convex sets, bounded or unbounded, including triangles, regular polygons, subsets of balls, right cylinders and cones, unbounded planar closed convex sets, etc. Then, we investigate the Γ-starshaped sets, and provide some conditions for a fan, a spherical sector and a right cylinder to be Γ-starshaped. Finally, we study the Γ-triple-convexity, which is a discrete generalization of Γ-convexity, and provide characterizations for all the 4-point sets, some 5-point sets and ℤd to be Γ-triple-convex.
