We consider a set $A$. $A$ is called convex if for every $x,y\in A$, we have the line segment $xy$ is also in $A$.
I want to generalize this notion, such that instead of one line segment, there can be $n$ line segment, where $n$ is some fixed number. Formally, there exist $x=a_1,a_2,\ldots,a_{n-1},a_n=y$, such that all the segments $a_1a_2$, $a_2a_3$,...,$a_{n-1}a_n$ is in $A$.
Is there a name for such sets?