The following problem is inspired by a problem in Pugh's Mathematical Analysis book. (Chapter 2 Problem 42).

In the problem he asks one to consider the standard Cantor set on the unit interval, and then to map it to the unit circle using the standard mapping $z \mapsto e^{2 \pi i z}$. Then given this set $D \subset S^1$ one can consider the subset of chords between all points in $D$. Namely $Chords(D) = \{t x + (1-t)y ~|~ x, y \in D, t \in [0,1]\}$. He asks if this set is compact and/or convex, to which the answers are yes and no respectively.

A further question can be asked: Does this set have measure zero? I believe naive bounds would show that if one constructs a cantor set by removing over $1/2$ of the interval at each iteration we indeed have the chords have measure 0. I am unsure of the result in the standard case of removing middle thirds.