TITLE: Is a certain subset of the disc a convex set?
QUESTION [12 upvotes]: Some one asked me this question and I thought about it and I don't have any good idea to solve that. Can some one help me and give me an idea to start solve that?
Draw a Cantor set $C$ on the circle and consider the set $A$ of all chords between points of $C$. Is $A$ a convex set?

REPLY [2 votes]: Start with a 3 point set {a,b,c} on the unit circle. The union of the 3 chords forms not a convex set. Blow up the points into 3 very small Cantor sets. The union of all the new chords still fails miserably to be a convex planar set. For example the origin is not contained in any chord, despite being in the convex hull of the set in question.