An example of a bounded $\mathsf C$-convex domain which is not biholomorphic to a convex domain - Nikolov, Nikolai; Pflug, Peter; Zwonek, Włodzimierz (Mathematica Scandinavica, 2008)
We show that the symmetrized bidisc is a $\mathsf C$-convex domain. This provides an example of a bounded $\mathsf C$-convex domain which cannot be exhausted by domains biholomorphic to convex domains.