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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?

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Piecewise connected? I'm not sure exactly what you mean, since if you have one line segment, then just partition your line segment to get a bunch of line segments. – Euler....IS_ALIVE Oct 16 '12 at 0:11

Assuming you allow arbitrarily large $n$, I believe the term is "polygonally connected".

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Aha, I should have state $n$ is fixed. I just fixed(heh) my question. – Chao Xu Oct 16 '12 at 2:06
up vote -4 down vote accepted

It is called the link diameter. So those sets can be called as sets with link diameter k.

An efficient algorithm for link-distance problems

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