Geometric properties of the tetrablock

In this short note we show that the tetrablock is i $\C$-convex domain. In the proof of this fact a new class of ($\C$-convex) domains is studied. The domains are natural caniddates to study on them the behavior of holomorphically invariant functions.


Introduction
Recently, two domains: symmetrized bidisc and tetrablock, arising from the µ-synthesis, turned out to be interesting examples in the geometric function theory. In particular, both domains are non-convex and even more, they cannot be exhausted by domains biholomorphic to convex ones and yet the Lempert function and the Carathéodory distance coincide on them (see [2], [5], [6], [7]). However, we have a more detailed knowledge on geometric properties in the case of the symmetrized bidisc. In particular, it is known that the symmetrized bidisc is C-convex and may be exhausted by strongly linearly convex domains (see [13] and [14]). All these facts show the importance of the domains from the point of view of the Lempert theorem on the equality of the Lempert function and the Carathéodory distance (see two papers of Lempert: [11] and [12]). We shall deal with analoguous properties of the tetrablock. More precisely, we show that the tetrablock is C-convex (see Corollary 4.2) which corrects the claim stated in [10] where, due to the typing error made in the formula describing the tetrablock, the converse was claimed.
It is interesting that the study of geometric properties of the tetrablock can be reduced to considering domains being generalizations of the symmetrized bidisc which may lead in the future to the study of other domains arising in the process of symmetrization of a 'nice' pseudoconvex complete Reinhardt domain. The domains G 2,ρ , which are Cconvex and aproximate the symmetrized bidisc, are natural candidates for the further study on the equality between the Lempert function and the Carathéodory distance as well as on the possibility of exhausting them with strongly linearly convex domains.
Basic notions, definitions and properties from the theory of invariant functions, linearly and C-convex domains that we shall use in the paper may be found in [9], [3] and [8].

Preliminary results
Below we present analytic definitions of both domains that will be of interest to us.
At first we shall prove that for any x ∈ E there is a hyperplane passing through x and omitting E. Thus we show the following.
As we shall see a more refined procedure than the one described above will lead us to the precise description of supporting hyperplanes which will finally lead to the proof of C-convexity of E. But before that we need some notations and auxiliary results.

Two-dimensional symmetrized domains characterizing the tetrablock
Motivated by Lemma 2.1 we shall see that the symmetrized images of special domains of the form where ρ ∈ (0, 1] will play a special role in the study of the geometry of E. Let us denote Then it follows from Lemma 2.1 that Φ ω (E) ⊂ G 2,|ω| , 0 < |ω| ≤ 1. We shall see that we even have the equality.
A domain D ⊂ C n is called C-convex if for any affine complex line l such that l ∩ D = ∅ the set l ∩ D is connected and simply connected.
For a domain D ⊂ C n and a point a ∈ C n we denote by Γ D (a) the set of all complex hyperplanes L such that (a + L) ∩ D = ∅. We shall often understand this set as the subset of P n−1 : Recall the basic criterion on C-convexity that we shall use: the bounded domain D ⊂ C n , n > 1, is C-convex iff for an x ∈ ∂D the set Γ D (x) is non-empty and connected (cf. e. g. Theorem 2.5.2 in [3]).
To show the other implication let L be such that L−x ∈ Γ E (x). Let L be given by the equation ay 1 + ωay 2 + ωcy 3 = d which may be written as a(y 1 + ωy 2 ) + cωy 3 = d. Then from the equality Φ ω (E) = G 2,|ω| (Lemma 2.2) we get that the line given by the equality as + cp = 0 belongs to Γ G 2,|ω| (Φ ω (x)).   Remark. Smoothness of the boundary point x together with the linear convexity means that Γ E (x) is a singleton. Note also that it follows from the earlier remark that the cases of boundary points considered in Theorem 3.3 (i. e. (r, r, 1) and (1, r, r)) represent all (up to linear automorphisms of E) non-smooth boundary points and it means that Theorem 3.3 gives a complete description of Γ E .
To show the other inclusion we proceed similarly as in the first case. Let L be such that L − (1, r, r) ∈ Γ E (1, r, r). Let L be given by the equation ay 1 + by 2 + cy 3 = d. First note that b = ωa for some |ω| ≤ 1.
In view of the above results we see that crucial for the proof of Cconvexity of the tetrablock is to find the description of Γ G 2,ρ . Recall that G 2 = G 2,1 is C-convex (see [13]). We have the following.
Moreover, if (s, p) = π 2 (λ 1 , λ 2 ) ∈ ∂G 2,ρ then for any ρ ∈ (0, 1) we have Proof. Let ρ ∈ (0, 1). The equalities in the first two cases follow from earlier remarks and the fact that the points considered there are smooth. Let us consider the third case. Note that any complex line passing through (s, p) and through a point from G 2,ρ must contain a point from the set π(D ρ ∩ {(µ 1 , µ) : ρ < |µ 1 |}) which implies that the set of points not in the set considered are the points of the form The last may be given in the form Since the function z → 1/(1 − z) maps the unit disc to {Re z > 1/2} we easily get that for the fixed µ 1 the set of points of the previous form for all µ with |µ 1 ||µ| < ρ is an open half plane. Now the set of numbers of the set in the theorem is the intersection of the complements of the sets of the last form. This implies that it is convex. We already know that [(0, 1)] ∈ Γ G 2,ρ (s, p) which finishes the proof. The fact that all sets Γ G 2,ρ (x), x ∈ ∂G 2,ρ , are non-empty and connected implies the C-convexity of G 2,ρ .
Proof. Linear convexity of E implies that in the case of a smooth boundary point x ∈ ∂E the set Γ E (x) is a singleton. Consider then the non-smooth point x ∈ ∂E. It is sufficient to consider the cases • x = (r, r, 1), r ∈ [0, 1], • x = (1, r, r), r ∈ [0, 1). Theorem 3.3 together with Theorem 4.1 (we also need to know that Γ G 2 (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. The fact that Γ E (1, r, r) is connected follows immediately from its description. This finishes the proof.