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I have been solving a problem that goes as follows:

If $F$ is a family of $n$ line segments on parallel lines and each triple can be intersected with a line then there is a line intersecting all the segments.

I solved it using Helly's theorem and everything was fine. But then I found a counterexample for the case when I don't have such a nice properties of segments in $F$. So I think that this would mean that the set of lines going through a segment is not always convex (otherwise I could apply the theorem). So I am wondering if there is a good characterization of the segments for which it is convex? For example, if a segment is on a line through the origin, is it necessarily convex? In general I am wondering to which families of segments can I apply Helly theorem in this way?

Jova
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  • Are you aware of that a user called “@klop” has posted this question this question a few days before you do yours? That question has received a comment from a user called “@Prism”. In this comment, he/she has mentioned about a theorem called $Theorem\space of\space Rey,\space Pastor\space and\space Santalo$. We think it is worth having a look. – YNK Dec 17 '22 at 13:28

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