关键词: $\Gamma$-convexity, $\Gamma$-starshaped sets, $\Gamma$-triple-convexity

Abstract: Let $\mathcal{F}$ be a family of sets in $\mathbb{R}^d\ (\mathrm{always}\ d\geq 2)$. A set $M\subset\mathbb{R}^d$ is called $\mathcal{F}$-convex, if for any pair of distinct points $x, y \in M$, there is a set $F\in \mathcal{F}$ such that $x, y \in F$ and $F \subset M$. We obtain the $\Gamma$-convexity, when $\mathcal{F}$ consists of $\Gamma$-paths. A $\Gamma$-path is the union of both shorter sides of an isosceles right triangle. In this paper we first characterize some $\Gamma$-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 $\Gamma$-starshaped sets, and provide some conditions for a fan, a spherical sector and a right cylinder to be $\Gamma$-starshaped. Finally, we study the $\Gamma$-triple-convexity, which is a discrete generalization of $\Gamma$-convexity, and provide characterizations for all the 4-point sets, some 5-point sets and $\mathbb{Z}^{d}$ to be $\Gamma$-triple-convex.

Key words: $\Gamma$-convexity, $\Gamma$-starshaped sets, $\Gamma$-triple-convexity