Abstract
Estimates for invariant distances of convexifiable, \(\mathbb{C }\)-convexifiable and planar domains are given.
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1 Introduction and results
Diederich and Ohsawa [6, p. 182] asked if \(D\) is a smooth bounded pseudoconvex domain in \({\mathbb{C }}^{n}\), then the following lower bound for the Bergman distance \(b_{D}\) holds: For fixed \(z\) and \(w\) close to \(\partial D\), one has that
where \(d_{D}(w)=\text{ dist }(w,\partial D)\) and \(c>0\) is a constant depending only on \(D.\) Błocki [4, Theorem 1.3] mentioned this fact for bounded convexifiable domains (not necessarily smooth).
We shall prove the estimate in the case of bounded \(\mathbb{C }\)-convex domains (or, more generally, \(\mathbb{C }\)-convexifiable). Recall that a set in \(\mathbb{C }^{n}\) is called \(\mathbb{C }\)-convex if all its intersections with complex lines are contractible (cf. [2, p. 25]). Note that a \(C^{1}\)-smooth bounded domain is \(\mathbb{C }\)-convex if and only the complex tangent hyperplane through any boundary point does not intersect the domain (cf. [2, Theorem 2.5.2]).
Let \(D\) be a domain in \(\mathbb{C }^{n}\). Denote by \(c_{D}\) and \(l_{D}\) the Carathéodory distance and the Lempert function of \(D,\) respectively:
where \(\mathbb{D }\) is the unit disk (we refer to [10] for basic properties of the objects under consideration). The Kobayashi distance \(k_{D}\) is the largest pseudodistance not exceeding \(l_{D}\). We have that
(if \(b_{D}\) is well-defined). Note also that \(k_{D}=l_{D}\) for any planar domain \(D\) (cf. [10, Remark 3.3.8(e)]). By Lempert’s theorem [11, Theorem 1], combining with a result by Jacquet [9, Theorem 5], \(c_{D}=l_{D}\) on any \(C^{2}\)-smooth bounded \(\mathbb{C }\)-convex domain \(D\) and hence on any convex domain. On the other hand, it follows by [14, Theorem 12] that there exists a constant \(c_{n}>0,\) depending only on \(n\), such that
for any \(\mathbb{C }\)-convex domain \(D\) in \(\mathbb{C }^{n}\), containing no complex lines (then \(b_{D}\) is well-defined). In other words, to estimate \(b_{D}\), it is enough to find lower bounds for \(c_{D}\) and upper bounds for \(l_{D}\).
Recall that \(b_{D}\) is the integrated form of Bergman metric
where
and
is the Bergman kernel on the diagonal (\(K_{D}(z)>0\) is assumed). So,
where the infimum is taken over all smooth curves \(\gamma :[0,1]\rightarrow D\) with \(\gamma (0)=z\) and \(\gamma (1)=w\).
Estimates for invariant distances of strictly pseudoconvex domains in \(\mathbb{C }^{n}\) and pseudoconvex domains of finite type in \(\mathbb{C }^{2}\) can be found in [3] (see also [1, 10]) and [8], respectively.
Recall now in details two estimates. The proof of [4, Theorem 5.4] (cf. also [12, Proposition 2.4]) implies that if \(D\) is a proper convex domain in \(\mathbb{C }^{n}\), then
(this proof uses only the existence of an appropriate supporting (real) hyperplane and the formula for the Poincaré distance of the upper half-plane). On the other hand, by [13, Theorem 1], for any \(C^{1+\varepsilon }\)-smooth bounded domain, there exists a constant \(c>0\) such that
(see [7, Proposition 2.5] for a stronger estimate for \(k_{D}\)).
The smoothness is essential as an example of a \(C^{1}\)-smooth bounded \(\mathbb{C }\)-convex planar domain shows (see [13, Example 2]). Moreover, using [16, p. 146, Theorem 7], one may find a bounded \(\mathbb{C }\)-convex planar domain for which there is no similar estimate with any constant instead of \(-1/2.\)
So, it natural to find an upper bound for \(l_{D}\) in the convex case and a lower bound for \(c_{D}\) in the \(\mathbb{C }\)-convex case.
Proposition 1
Let \(D\) be a proper convex domain in \(\mathbb{C }^{n}\). Then
Footnote 1 In particular, if, in addition, \(D\) is bounded, then for any compact subset \(K\) of \(D\), there is a constant \(c_{K}>0\) such that
The last estimate for \(k_{D}\) instead of \(b_{D}\) (and \(K\) a singleton) is the content of [12, Proposition 2.3]. Similar estimates for the Kobayashi distance of pseudoconvex Reinhardt domains can be found in [19].
Proposition 2
Let \(D\) be a proper \(\mathbb{C }\)-convex domain in \(\mathbb{C }^{n}\). Then
Hence, if, in addition, \(D\) is bounded, then for any compact subset \(K\) of \(D\), there is a constant \(c_{K}>0\) such that
Note that by [5, p. 2381], the first estimate in Proposition 2 implies the following
Corollary 3
The Bergman and Szegö kernels (on the diagonal) are comparable on any \(C^{2}\)-smooth bounded \(\mathbb{C }\)-convex domain.
We point out that [5, Theorem 1.3] deals with the convex case.
Remark
-
(a)
The estimate for \(l_{D}\) is sharp when \(z\rightarrow w\). Moreover, it is sharp up to a constant when \(z\) is fixed and \(w\rightarrow \partial D\). Indeed, denote by \(R_{D}(z,w)\), the right-hand side of the first inequality in Proposition 1. If \(\theta \in (0,\pi )\) and \(D_{\theta }=\{z\in \mathbb{C }_{*}:|\arg z|<\theta \}\), then
$$\begin{aligned} \lim _{\theta \rightarrow 0}\lim _{x\rightarrow 0+}\frac{l_{D_\theta }(1,x)}{R_{D_\theta }(1,x)}=\frac{\pi }{4}. \end{aligned}$$ -
(b)
The factor \(1/4\) in the bound for \(c_{D}\) is optimal as \(D=\mathbb{C }_{*}\setminus {\mathbb{R }}^{+}\) shows.
-
(c)
Estimates for the infinitesimal forms of the distances under consideration, namely, the Carathéodory, Kobayashi and Bergman metrics, of convex and \(\mathbb{C }\)-convex domains can be found in [14]. The bounds there depend only on the distance to the boundary from the respective point in the respective direction.
Our main result is in the spirit of [4, Theorem 1.3], where a lower bound for the Bergman metric is mentioned in the locally convexifiable case (and a hint for a proof is given).
Proposition 4
Let \(D\) be a bounded domain in \(\mathbb{C }^{n}\) which is locally \(\mathbb{C }\)-convexifiable, i.e., for any point \(a\in \partial D\), there exist a neighborhood \(U_{a}\) of \(a\), an open set \(V_{a}\) in \(\mathbb{C }^{n}\) and a biholomorphism \(F_{a}:U_{a}\rightarrow V_{a}\) such that \(F_{a}(D\cap U_{a})\) is \(\mathbb{C }\)-convex. Then, there exists a constant \(c>0\) such that for any compact subset \(K\) of \(D\) one can find a constant \(c_{K}>0\) with
where \(s_{D}=k_{D}\) or \(s_{D}=b_{D}\).
Moreover, if \(D\) is locally convexifiable or \(C^{1+\varepsilon }\)-smooth and locally \(\mathbb{C }\)-confexifiable, then for any compact subset \(K\) of \(D\), one can find a constant \(c^{\prime }_{K}>0\) with
Finally, we consider the planar case. We shall say that a boundary point \(p\) of a planar domain \(D\) is Dini-smooth if \(\partial D\) near \(p\) is a Dini-smooth curve \(\gamma :[0,1]\rightarrow \mathbb{C }\).Footnote 2 Call a planar domain Dini-smooth if it is Dini-smooth near any boundary point.
Proposition 5
Let \(p\) be a Dini-smooth boundary point of a planar domain \(D\). Then, for any neighborhood \(U\) of \(p\) and any compact subset \(K\) of \(D\), there exist a neighborhood \(V\) of \(p\) and a constant \(c>0\) such that
where \(s_{D}=c_{D},\, s_{D}=l_{D}(=k_{D})\) or \(s_{D}=b_{D}/\sqrt{2}\).
Since \(k_{D}\) and \(b_{D}\) are the integrated forms of \(\kappa _{D}\) and \(\beta _{D}\), we get the following
Corollary 6
Let \(p\) and \(q\) be different Dini-smooth boundary points of a planar domain \(D\). If \(s_{D}=l_{D}(=k_{D})\) or \(s_{D}=b_{D}/\sqrt{2}\), then the function
is bounded for \(z\) near \(q\) and \(w\) near \(p\).
In general, \(c_{D}\) is not an inner distance (even in the plane). So, the next proposition is not a direct consequence of Proposition 5.
Proposition 7
Let \(p\) and \(q\) be different Dini-smooth boundary points of a planar domain \(D\). Then, the function
is bounded for \(z\) near \(q\) and \(w\) near \(p\).
The next result is optimal for the boundary behavior of \(c_{D}\) and \(l_{D}(=k_{D})\) in the planar case. It is more general than the last results, but its proof uses these results. Similar (and slightly weaker) result for \(k_{D}\) on \(C^{2}\)-smooth strictly pseudoconvex bounded follows by [3, Theorem 1, Proposition 1.2].
Proposition 8
Let \(D\) be a Dini-smooth bounded planar domain.Footnote 3 Then, there exists a constant \(c\ge 1\) such that
In particular, the function \(l_{D}-c_{D}\) is bounded on \(D\times D\).
It is shown in [18, Theorem 1] that if \(D\) is strongly pseudoconvex domain in \(\mathbb{C }^{n}\), then
We have the following planar extension of this result.
Proposition 9
If \(D\) is finitely connected bounded planar domain without isolated boundary points, then
2 Proofs
Proof of Proposition 1
Denote by \(C_{z,w}\) the convex hull of the union of the disks \(\mathbb{D }(z,d_{D}(z))\) and \(\mathbb{D }(w,d_{D}(w)),\) lying in the complex line through \(z\) and \(w\). Let \(\gamma (t)=z+t(w-z)\). Since \(C_{z,w}\subset D\) and \(l_{C_{z,w}}=k_{C_{z,w}}\) is the integrated form of the Kobayashi metric \(\kappa _{C_{z,w}}\), Footnote 4 then
This inequality and (1) lead to the wanted result for \(b_{D}\).
Proof of Proposition 2
Let \(p(w)\in \partial D\) be such that \(||w-p(w)||=d_{D}(w)\). Since \(E\) is \({\mathbb{C }}\)-convex, there exists a hyperplane \(H_{p(w)}\) through \(p(w)\) and disjoint from \(D\) (cf. [2, Theorem 2.3.9(ii)]). Denote by \(D_{w}\) and \(z_{w}\) the projections of \(D\) and \(z\) onto the complex line through \(w\) and \(p(w)\) in direction \(H_{(p(w)},\) respectively. By [2, Theorem 2.3.6], \(D_{w}\) is a simply connected domain and \(p(w)\in \partial D_{w}\). Denote by \(\psi _{w}\in \mathcal{O }(\mathbb{D },D_{w})\) a Riemann map such that \(\psi _{w}(0)=z_{w}\). If \(\psi _{w}(\alpha _{w})=w\), then
By [16, p. 139, Corollary 6] (which is a consequence of the Köbe 1/4 and the Köbe distortion theorems),
Since \(d_{D_{w}}(w)=d_{D}(w)\) and \(|\psi {^{\prime }}_{w}(0)|\ge d_{D_{w}}(z_{w})\ge d_{D}(z),\) it follows that
This inequality and \(b_{D}\ge c_{D}\) imply the desired result for \(b_{D}\).
Proof of Proposition 4
Footnote 5 First, we shall prove the lower bound.
Note that
Then, by Proposition 2, we may find a finite set \(M\subset \partial D\) and a constant \(c_{1}>0\) such that
where \(V_{a}\subset U_{a}\) is a neighborhood of \(a\) such that \(\partial D\subset \cup _{a\in M} V_{a}\).
Denote now by \(S_{D}\) the Kobayashi or Bergman metrics of \(D\). By localization principles (cf. [10, Proposition 7.2.9 and Proposition 6.3.5], since \(D\) is pseudoconvex), there exists a constant \(c_{2}>0\) such that
Let \(W_{a}\Subset V_{a}\) be such that \(W=\cup _{a\in M} W_{a}\) does not intersect \(K\) and contains \(\partial D\). Set \(r=\min _{a\in M}\text{ dist }(\partial W_{a},\partial V_{a})\).
Let \(\varepsilon >0\). Since \(s_{D}\) is the integrated form of \(S_{D}\), for any \(z\in K\) and \(w\in D\cap W\), there exists a smooth curve \(\gamma :[0,1]\rightarrow D\) with \(\gamma (0)=z,\,\gamma (1)=w\) and
Let \(t_{1}=\max \{t\in (0,1):\gamma (t)\in G=D\setminus W\}\). Choose a point \(a_{1}\in M\) such that \(\mathbb{B }_{n}(\gamma (t_{1}),r)\subset V_{a_{1}}\). Let \(t_{2}=\sup \{t\in (t_{1},1]:\gamma ([t_{1},t))\in V_{a_{1}}\}\) and etc. In this way, we may find numbers \(0<t_{1}<\dots <t_{N+1}=1\) and points \(a_{1},\dots ,a_{N+1}\in M\) such that \(\gamma [t_{j},t_{j+1})\subset D\cap V_{a_{j}}\) and \(||\gamma (t_{j+1})-\gamma (t_{j})||\ge r,\,1\le j\le N\). Then
where \(c_{3}=4c_{1}c_{2}\).
On the other hand, since \(D\) is a bounded domain, there exists a constant \(c_{4}>0\) such that \(s_{D}(z_{1},z_{2})\ge c_{4}||z_{1}-z_{2}||\). Then
So,
The case when \(w\in G\) is trivial which completes the proof of the lower bound.
The proof of the upper bound is easier. Fix a point \(a\in \partial D\). It is enough to find a constant \(c^{\prime }_{a,K}>0\) such that the estimate holds for \(w\) near \(a\). Take a point \(u\in U_{a}\) and a neighborhood \(V_{a}\Subset U_{a}\) of \(a\) and a point \(u\in D\cap U_{a}\). Proposition 1, (3) and (4) imply that
The upper bound for \(b_{D}\) follows similarly. It suffices to use that
in view of [10, Proposition 6.3.5] and (1).
Proof of Proposition 5 for
\(c_{D}\) and \(l_{D}\) We may find a Dini-smooth Jordan curve \(\zeta \) such that \(\zeta =\partial D\) near \(p\) and \(D\subset G:=\zeta _{\text{ ext }}\). Take a point \(a\not \in \overline{G}\) and consider the union \(G_{e}\) of \(0\) and the image of \(G\) under the map \(\varphi :z\rightarrow (z-a)^{-1}\). There exists a conformal map \(\psi :G_{e}\rightarrow \mathbb{D }\). It extends to a \(C^{1}\)-diffeomorphism from \(\overline{G_{e}}\) to \(\overline{\mathbb{D }}\) (cf. [20, Theorems 3.5]). Setting \(\eta =\psi \circ \varphi \), then
Now the lower bound for \(c_{D}\) follows by the same bound for \(c_{\mathbb{D }}\) and an inequality of type (4).
The estimate
follows by (3). It can be also obtained in the following way. There exist a Dini-smooth domain simply connected domain \(G_{i}\subset D\) and a neighborhood \(V\) of \(p\) such that \(\partial G\cap V=\partial D\cap V\). Take a point \(u\in V\). Since \(l_{D}=k_{D}\), then
It remains to repeat the final arguments from the first paragraph.
Proof of Proposition 5 for
\(b_{D}\) Footnote 6 Choosing \(G\) as above, then
By the Dini-smoothness,
We may assume that \(\eta (p)=1\). So, it is enough to get the estimates for \(D\subset \mathbb{D }\) such that \(F=\mathbb{D }\cap \mathbb{D }(1,r)\subset D\) for some \(r\in (0,1)\).
First, we shall prove that if \(0<r^{\prime }<r\), then
for some constant \(c^{\prime }>0\).
For a domain \({\Omega }\subset \mathbb{C }\) set \(\beta _{{\Omega }}(z)=B_{{\Omega }}(z;1)\) and \(\kappa _{{\Omega }}(z)=\kappa _{{\Omega }}(z;1)\). Let \({\check{F}}=\mathbb{D }\setminus F\) and
Then, for any \(r^{\prime \prime }\in (r^{\prime },r)\), we may find a constant \({\tilde{c}}>0\) such that
(for the equality use that \(F\) is biholomorphic to \(\mathbb{D }\) and for the inequality “between the lines” cf. [10, Proposition 7.2.9]).
Let \(z\in K, w\in F^{\prime }\) and \(w^{\prime }=[0,w]\cap \partial D(1,r^{\prime \prime })\). Then
for some constant \(c^{\prime }>0\).
Now, shrinking \(r\) such that \(\mathbb{D }(1,r)\subset U\), it remains to prove that
for some constant \(c^{\prime \prime }>0\).
We have that
For \(z\in {\check{F}}, w\in F^{\prime }\), and \(\varepsilon >0\), there exists a smooth curve \(\gamma :[0,1]\rightarrow D\) with
Let \(t_{0}=\sup \{t\in (0,1):\gamma (t)\not \in F^{\prime \prime }\}\). Then,
where \(\hat{b}_{\mathbb{D }}\) is the integrated form of the Finsler pseudometric
It remains to use that, shrinking \(r^{\prime }\) (if necessary),
for some constant \(c^{\prime \prime }>0\) (cf. [3, Theorem 1.1]).
Proof of Corollary 6
Since \(k_{D}\) and \(b_{D}\) are the integrated forms of \(\kappa _{D}\) and \(\beta _{D}\), the boundedness from below follows by the first inequality in Proposition 5 (cf. the proof of [10, Proposition 10.2.6]). Choosing a point \(a\in D\), the boundedness from above is a consequence of the inequality \(s_{D}(z,w)\le s_{D}(z,a)+s_{D}(a,w)\) and the second inequality in Proposition 5.
Proof of Proposition 7
In virtue of the inequality \(c_{D}\le k_{D}\) and Corollary 6, we have to prove only the boundedness from below. For this, take disjoint Dini-smooth Jordan curves \(\zeta ^{\prime }\) and \(\zeta ^{\prime \prime }\) such that \(\zeta ^{\prime }=\partial D\) near \(p, \zeta ^{\prime \prime }=\partial D\) near \(q\) and \(D\subset G:=\zeta ^{\prime }_{\text{ ext }}\cap \zeta ^{\prime \prime }_{\text{ ext }}\). Note that any Dini-smooth bounded double-connected planar \({\tilde{G}}\) domain can be conformally map to some annulus \(A_{r}=\{z\in \mathbb{C }:1/r<|z|<r\} (r>1)\), and the respective mapping extends to a \(C^{1}\)-diffeomorphism from \(\overline{\tilde{G}}\) to \(\overline{A_{r}}\).Footnote 7
Then, proceeding similarly to the proof of Proposition 5 for \(c_{D}\), it is enough to show that
is bounded from below for \(z\in \mathbb{R }\) near \(r\) and \(w\) near \(p\), where \(|p|=1/r\); this is equivalent to
for some constant \(c>0\).
Recall that (cf. [10, Proposition 5.5])
where \(f\) is a holomorphic function on \(\overline{A_{r}\times A_{r}}\setminus \{u=v\in \partial A_{r}\}\) and \(|f(u,v)|=1\) if \(|u|=r, v\in \overline{A_{r}}\) or \(u\in \overline{A_{r}}, |v|=1/r (u\ne v)\).
In particular,
at any point \((u,v)\) with \(|u|=r\) and \(|v|=1/r\). Then, by the Taylor expansion,
This implies that
(the constant can be chosen the same for \(z\) near \(r\) and \(w\) away from \(r\)). Since \(f(r,\cdot )\) is a unimodular constant and \(d_{A_{r}}(w)=d_{A_{r}}(|w|)\), it follows that
Further, \(c_{A_{r}}(z,|w|)=c_{A_{r}}(z,t)+c_{A_{r}}(t,|w|)\) for \(t\in [|w|,z]\) (cf. [10, Lemma 5.11(b)]). Then, Proposition 5 implies that
Hence we may choose \(c=c_{2}+c_{3}\) which completes the proof.
Proof of Proposition 8
Using Corollary 6 and Proposition 7, it is enough to prove the inequalities for \(z\) and \(w\) near a fixed point \(p\in \partial D\). Moreover, it is easy to see that these inequalities are equivalent to
for some constant \(c\ge 1\).Footnote 8
To prove the lower bound for \(\tanh c_{D}(z,w)\), let \(\eta \) be as in the proof of Proposition 5 for \(c_{D}\) and \(l_{D}\). Then, it is not difficult to find a constant \(c_{1}>0\) such that
where \(z_{1}=\eta (z)\) and \(w_{1}=\eta (w)\). It remains to use that, similarly to (4), \(d_{D}\ge c_{2} d_{\mathbb{D }}\) and \(|z_{1}-w_{1}|\ge c_{2}|z-w|\) for some constant \(c_{2}>0\).
The proof of the upper bound for \(\tanh l_{D}(z,w)\) is similar (by using \(G_{i}\) from the second part of the proof mentioned above) and we skip it.
Proof of Proposition 9
By the Köbe uniformization theorem, we may assume that \(\partial D\) consists of disjoint circles. Using Proposition 8 and compactness, it is enough to prove that for any point \(p\in \partial D\),
Applying an inversion, we may suppose that the outer boundary of \(D\) is the unit circle \(\Gamma \) and \(p\in \Gamma \). Let \(U\) be a disk centered at \(p\) such that \(\mathbb{D }\cap U\subset D\). Then,
Considering \(\mathbb{D }\) as a part of the unit ball in \(\mathbb{C }^{2}\), it follows that the last ratio tends to 1 as a particular case of the same result for strongly pseudoconvex domains (see [18, Proposition 3]).
Notes
If \(d(z)=d(w),\) then \(l_D(z,w)\le ||z-w||/d(w).\)
This means that \(\int _{0}^{1}\frac{\omega (t)}{t}dt<\infty \), where \(\omega \) is the modulus of continuity of \(\gamma ^{\prime }\).
This means that \(D\) is Dini-smooth near any boundary point.
If \(D\subset \mathbb{C }^{n}\), then \(\kappa _{D}(z;X)=\inf \{|\alpha |:\exists \varphi \in \mathcal{O }(\mathbb{D },D) \text{ with } \varphi (0)=z,\alpha \varphi ^{\prime }(0)=X\}.\)
Some difficulty arises from the fact that, in contrast to invariant metrics, general localization principles for invariant distances are not known. However, a strong localization principle holds for \(k_{D}\) and \(c_{D}\) if \(D\) is strongly pseudoconvex (see [18, Proposition 3, Theorem 1].
We have to modify the previous proof, since the Bergman distance is not monotone under inclusion of planar domains; to see this, use [15, Example 7].
To see this, we can proceed as follows (S. R. Bell, private communication). First, take a conformal mapping \(\varphi _{1}\) from the domain bounded by the outer boundary of \({\tilde{G}}\) to \(\mathbb{D }\). Next, choose a point \(a\) in the interior of the inner boundary \(\Gamma \) of \(\psi _{1}({\tilde{G}})\) and set \(\psi _{2}:z\rightarrow (z-a)^{-1}\). Consider now a conformal mapping \(\psi _{3}\) from the domain bounded by \(\psi _{2}(\Gamma )\) to \(\mathbb{D }\). Then, \(\psi =\psi _{3}\circ \psi _{2}\circ \psi _{1}\) maps conformally \({\tilde{G}}\) to a bounded double-connected planar domain \(G^{\prime }\) with real-analytic boundary. It remains to apply the reflection principle to a conformal mapping from \(G^{\prime }\) to \(A_{r}\).
These estimates imply the bounds for the Green function \(g_{D}\) from the crucial Lemma 4.2 in [17], since \(\tanh c_{D}\le \exp (-2\pi g_{D})\le \tanh l_{D}\).
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The author would like to thank P. Pflug, T. Warszawski and the referee for their remarks on the paper.
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Nikolov, N. Estimates of invariant distances on “convex” domains. Annali di Matematica 193, 1595–1605 (2014). https://doi.org/10.1007/s10231-013-0345-7
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DOI: https://doi.org/10.1007/s10231-013-0345-7
Keywords
- Carathéodory
- Kobayashi and Bergman distances
- Bergman and Szegö kernels
- Convex
- Convexifiable and \(\mathbb{C }\)-convex domains