An example of a bounded $\Bbb C$-convex domain which is not   biholomoprhic to a convex domain
  We show that the symmetrized bidisc is a $\Bbb C$-convex domain. This provides an example of a bounded $\Bbb C$-convex domain which cannot be exhausted by domains biholomorphic to convex domains. 
Nikolai Nikolov, Peter Pflug and Wlodzimierz Zwonek