Geometric properties of the tetrablock

In this short note, we show that the tetrablock is a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{C}}$$\end{document}-convex domain. In the proof of this fact, a new class of (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{C}}$$\end{document}-convex) domains is studied. The domains are natural candidates to study on them the behavior of holomorphically invariant functions.

in the process of symmetrization of a 'nice' pseudoconvex complete Reinhardt domain. The domains G 2,ρ , which are C-convex (see Theorem 4.1) and approximate 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 [3,8,9].

Preliminary results.
Below we present analytic definitions of both domains that will be of interest to us.
Recall that the tetrablock may be defined as follows (see [1]) and the symmetrized bidisc as follows (see e.g. [2]) Let us begin our study with presenting a close relation between E and G 2 , which may be a good starting point for us.
Actually, the equivalence of the first two properties follows from the following observation which holds for all |ω| = 1, together with the fact that the inequality above becomes an equality for some |ω| = 1. 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) : Remark. The equality (2.1) defining the tetrablock allows us to find out that any point x ∈ ∂E such that x 1 =x 2 x 3 and x 2 =x 1 x 3 is a smooth boundary point of ∂E. Moreover, the condition x ∈ ∂E and x 1 =x 2 x 3 or x 2 = x 1 x 3 means that x = (re iθ , re iτ , e i(θ+τ ) ) or x = (e iθ , re iτ , re i(θ+τ ) ) or x = (re iθ , e iτ , re i(θ+τ ) ) for some r ∈ [0, 1], θ, τ ∈ R. Composing with the automorphism of E given by the formula (e −iθ y 1 , e −iτ y 2 , e −i(θ+τ ) y 3 ), we shall often be able to reduce the problem to special cases: x = (r, r, 1) or x = (1, r, r) or x = (r, 1, r), r ∈ [0, 1].
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. Proof. Let x ∈ E. Making use of the description of E from Lemma 2.1 for such an x, we find an ω ∈ D with Φ ω (x) = (x 1 + ωx 2 , ωx 3 ) ∈ G 2 . 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 = ∅.
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 the 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 We shall see that we even have the equality. Proof. Actually, let ρ := |ω| and take (s, p) ∈ G 2,ρ . Put It easily follows from (2.1) that x ∈ E and Φ ω (x) = (s, p).
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 a 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]). One may also easily see that
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| (Proposition 3.1), 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. Actually in the other case a, b = 0 and a = ωb for some 0 < |ω| < 1, and then in view of 4. C-convexity of G 2,ρ and the tetrablock. 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.
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 164 W. Zwonek Arch. Math.
where ρ < |μ 1 | < 1, |μμ 1 | < ρ. 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 G2,ρ (s, p), which finishes the proof. The fact that all sets Γ G2,ρ (x), x ∈ ∂G 2,ρ , are non-empty and connected implies the C-convexity of G 2,ρ .

Corollary 4.2. E is C-convex.
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).