On the m-point convexity

Let $S\subset \mathbb{R}^d$ $(d\geq 2)$. A set $S$ is said to be $m$-point convex, if for every $m$ distinct points in $S$, at least one of the line-segments determined by them lies in $S$. We also say that $S$ has property $P_m$. Let ${x,y,z}\in \mathbb{R}^{d}$. If $\mathrm{conv}\{x,y,z\}$ is a right triangle, then $\{x,y,z\}$ is called a {\it right triple}. A set $S$ is said to have the right-$3$-point property,if, for every right triple of $S$, at least one of the line-segments determined by them belongs to $S$. In particular, it has the double right-$3$-point property, if, for every right triple in $S$, at least two of the line-segments determined by them belong to $S$. In this paper, we further investigate $m$-point convex sets and establish the relationship between the sets with the double right-$3$-point property and convex sets in $\mathbb{R}^d$.