Let S be a subset of a linear space. Let S1 be the union of all line segments that join pairs of points in S. Now what happens if we repeat this process and construct S2, S3,....(Thus for example S2 is the union of line segments in S1)?

My guess is that the end-result should be the convex hull of the set S, but I am not able to prove/disprove this.

Thank you,

Sanjeev