How many Steiner Symmetrizations does it take to make an arbitrary set convex?

Question

I have not seen this question investigated before but I might be wrong:

1. Can any subset of $\mathbb{R}^d$ be turned into a convex set by finitely many steiner symmetrizations?

2. If yes, is the number of symmetrizations necessarily bounded, i.e. does there exist a dimensional constant such that it will always take at most $m_d$ symmetrizations (this would be interesting)

3. If no, what is a counterexample, i.e. a set that can not be made convex within a finite number of symmetrizations?

I hope somebody knows something, this is super a interesting situation for me. Thanks for any answers.

Answer 1

Koch's snow flake needs an infinite number of Steiner symmetrizations.