An example of a bounded $\mathsf C$-convex domain which is not biholomorphic to a convex domain

Abstract

We show that the symmetrized bidisc is a C-convex domain. This provides an example of a bounded C-convex domain which cannot be exhausted by domains biholomorphic to convex domains.2000 Mathematics Subject Classification. 32F17.

Geometric properties of the tetrablock

“…Note that the hyperconvexity of G 2 (see e.g. [13]) together with the maximum principle for subharmonic functions gives the equivalence of the two latter conditions. Let us also denote Φ ω (x) := (…”

“…The linear convexity of G 2 (see e.g. [13]) implies that there is a line l = {(s, p) ∈ C 2 : as+bp = c} with Φ ω (x) ∈ l and l∩G 2 = ∅. Then x ∈ L := {y ∈ C 3 : ay 1 +aωy 2 +bωy 3 = c} and, as one may easily check, L ∩ E = ∅.…”

“…In view of the above results, we see that crucial for the proof of C-convexity of the tetrablock is to find the description of Γ G2,ρ . Recall that G 2 = G 2,1 is C-convex (see [13]). We have the following.…”

“…Theorem 3.3 together with Theorem 4.1 (we also need to know that Γ G2 (π(λ)) is connected and contains the point [(0, 1)]-this follows from [13]) imply that the set Γ E (r, r, 1) is the union of connected sets whose intersection is non-empty (it contains the point [(0, 0, 1)]), so it is connected too.…”

Geometric properties of domains related to μ-synthesis

“…[26]), there is an affine complex line L ⊂ C r 1 such that L ∩ G r 1 = ∅ and the set L ∩ G r 1 either is not connected or is not simply connected. Consequently, L := L × {0} N −r 1 ⊂ C N is an affine complex line such that L ∩ E E = ∅ and the set L ∩ E E either is not connected or is not simply connected.…”