C-convexity

convexity in complex analysis

A domain or compact subset $E$ in $\mathbf{C} ^ { n }$ is said to be $\mathbf{C}$-convex if for any complex line $\operatorname{l} \subset \mathbf{C} ^ { n }$ the intersection $E \cap \bf l$ is both connected and simply connected (meaning that its complement in the Riemann sphere $\text{l} \cup \{ \infty \}$ is connected; cf. also Connected set; Simply-connected domain). The notion of $\mathbf{C}$-convexity is an intermediate one, in the sense that any geometrically convex set is necessarily $\mathbf{C}$-convex, whereas $\mathbf{C}$-convexity implies holomorphic convexity (or pseudo-convexity; cf. also Pseudo-convex and pseudo-concave). In particular, a real subset $E$ in $\mathbf{R} ^ { n } \subset \mathbf{C} ^ { n }$ is $\mathbf{C}$-convex if and only if it is convex in the ordinary, geometrical, sense. Open or compact $\mathbf{C}$-convex subsets are in several respects the natural sets when it comes to studying properties of holomorphic functions that are invariant under affine (or projective) transformations. They often play a role analogous to that of convex sets in real analysis.

Topology of $\mathbf{C}$-convex sets.

Any $\mathbf{C}$-convex domain is homeomorphically equivalent to the open unit ball $| z _ { 1 } | ^ { 2 } + \ldots + | z _ { n } | ^ { 2 } < 1$, and a compact $\mathbf{C}$-convex set $E$ is also topologically simple in the sense that it has vanishing reduced cohomology:

\begin{equation*} H ^ { 0 } ( E ) = \mathbf{Z} , \quad H ^ { p } ( E ) = 0 , p > 0. \end{equation*}

The operations of forming the closure and the interior are not well-adapted to $\mathbf{C}$-convexity. There exist, e.g., compact $\mathbf{C}$-convex sets with non-connected interior. Also, the intersection of two $\mathbf{C}$-convex sets is not necessarily $\mathbf{C}$-convex.

Projective invariance.

It is natural to consider a $\mathbf{C}$-convex set $E \subset \mathbf{C} ^ { n }$ also as a subset of the complex projective space $\mathbf{P} ^ { n } \supset \mathbf{C} ^ { n }$. Each non-trivial complex linear mapping $\tilde{T} : {\bf C} ^ { m + 1 } \rightarrow {\bf C} ^ { n + 1 }$ descends to a mapping $T : \mathbf{P} ^ { m } \backslash X \rightarrow \mathbf{P} ^ { n }$, where $X$ corresponds to the kernel of $\tilde{T}$. The mapping $T$ is then called a projective transformation, and $\mathbf{C}$-convexity is invariant under any such transformation: If $E$ and $F$ are $\mathbf{C}$-convex subsets of ${\bf P} ^ { m } \backslash X$ and $\mathbf{P} ^ { n }$, respectively, then both $T ( E )$ and $T ^ { - 1 } ( F )$ are also $\mathbf{C}$-convex.

A $\mathbf{C}$-convex set $E \subset {\bf C} ^ { n } \subset {\bf P} ^ { n }$ is said to be non-degenerate if it is not of the form $T ( F )$ or $T ^ { - 1 } ( F )$, where $F \subset \mathbf{P} ^ { n - 1 }$ and $T$ is a projective transformation. Examples of degenerate $\mathbf{C}$-convex sets are those that are contained in complex hyperplanes, or of the form $E \times \mathbf C$. It is an interesting fact that far from all one-dimensional $\mathbf{C}$-convex sets can occur as the intersection of a multi-dimensional non-degenerate $\mathbf{C}$-convex set with a complex line.

Dual complement and linear convexity.

If $E$ is any subset of $\mathbf{C} ^ { n } \subset \mathbf{P} ^ { n }$, then its dual complement $E ^ { * }$ is, by definition, the collection of all complex hyperplanes that do not intersect $E$. When $n = 1$ this is just the usual complement of a set in the Riemann sphere, and in higher dimensions $E ^ { * }$ can be considered as a subset of the dual projective space ${\bf P}^ { n^* }$. If $E$ is an open (or compact) $\mathbf{C}$-convex set, then its dual complement $E ^ { * }$ is a compact (respectively, open) $\mathbf{C}$-convex set. Moreover, such a $\mathbf{C}$-convex set $E$ is also linearly convex in the sense that $E = E ^ { * * }$, i.e. the complement of $E$ is a union of complex hyperplanes. There are, however, many linearly convex sets that are not $\mathbf{C}$-convex. For instance, any Cartesian product of subsets of $\mathbf{C}$ is linearly convex, but it is $\mathbf{C}$-convex only if each factor is convex in the usual sense. A connected component of a linearly convex set $E$ is not necessarily linearly convex. In fact, a compact or open set $E$ which is equal to one or several connected components of its linearly convex hull $E ^ { * * }$ is said to be weakly linearly convex. For an open set $E$, weak linear convexity amounts to the condition that through any boundary point $a \in \partial E$ there should pass a complex hyperplane not intersecting $E$, and this does not in general imply linear convexity of $E$. Any weakly linearly convex open set $E$ in $\mathbf{C} ^ { n }$ is pseudo-convex.

Boundary properties.

If a weakly linearly open set in $\mathbf{P} ^ { n }$, $n > 1$, has a boundary of class $C ^ { 1 }$, then it is automatically $\mathbf{C}$-convex. In particular, for bounded domains in $\mathbf{C} ^ { n }$ with $C ^ { 1 }$ boundary, the notions of $\mathbf{C}$-convexity, linear convexity and weak linear convexity all coincide.

When the smoothness assumption is strengthened, so that $E = \{ z \in \mathbf C ^ { n } : \rho ( z ) < 0 \}$ is given by a defining function $\rho$ of class $C ^ { 2 }$, then one considers the quadratic form

\begin{equation*} H _ { \rho } ( a ; w ) = \end{equation*}

\begin{equation*} = 2 \operatorname { Re } \left( \sum _ { j ,\, k } \rho _ { j k } ( a ) w _ { j } w _ { k } \right) + 2 \sum _ { j ,\, k } \rho _ { j \overline { k } } ( a ) w _ { j } \overline { w } _ { k }, \end{equation*}

where $a \in \partial E$, , $\rho _ { j k } = \partial ^ { 2 } \rho / \partial z _ { j } \partial z _ { k }$, and $\rho _ { j \overline { k } } = \partial ^ { 2 } \rho / \partial z _ { j } \partial \overline{z} _ { k }$. This quadratic form is called the Hessian of $\rho$ at $a$, whereas its Hermitian part

\begin{equation*} L _ { \rho } ( a ; w ) = \sum _ { j , k } \rho _ { j \overline { k } } ( a ) w _ { j } \overline { w } _ { k } \end{equation*}

is called the Levi form (cf. also Hessian of a function). A smoothly bounded domain $E$ is convex if and only if the Hessian of its defining function is positive semi-definite when restricted to the real tangent plane at any $a \in \partial E$. Similarly, a domain $E$ is pseudo-convex precisely if the restriction of the Levi form to the complex tangent plane is positive semi-definite. The notion of $\mathbf{C}$-convexity lies in between: A domain $E \subset \mathbf{P} ^ { n }$ with boundary of class $C ^ { 2 }$ is $\mathbf{C}$-convex if and only if for any $a \in \partial E$ the Hessian is positive semi-definite on the complex tangent plane at $a$.

A complex hyperplane is said to be a tangent plane to an arbitrary open set $E \subset \mathbf{C} ^ { n }$ if it intersects the boundary $\partial E$ but not $E$ itself. When $n > 1$, a connected open set $E$ is $\mathbf{C}$-convex if and only if, for any $a \in \partial E$, the set of complex tangent planes to $E$ at $a$ is a non-empty connected subset of the dual complement $E ^ { * }$.

Fantappiè transform.

If $E$ is an open (or compact) subset of $\mathbf{C} ^ { n }$, one denotes by $A ( E )$ the vector space of holomorphic functions on $E$, endowed with the projective (respectively, inductive) limit topology. An element $\mu$ of the dual space $A ^ { \prime } ( E )$ is called an analytic functional on $E$. If $E$ contains the origin, then any hyperplane not intersecting $E$ is of the form $\{ z \in \mathbf{C} ^ { n } : 1 + \langle z , \zeta \rangle \neq 0 \}$, and the Fantappiè transform $\tilde{\pi}$ is defined to be the element of $A ( E ^ { * } )$ given by

\begin{equation*} \widetilde{\mu} ( \zeta ) = \mu \left( \frac { 1 } { ( 1 + \langle \cdot , \zeta \rangle ) } \right). \end{equation*}

Now, if $E$ is $\mathbf{C}$-convex, then the mapping $\mu \mapsto \tilde{\mu}$ provides a topological isomorphism between the spaces $A ^ { \prime } ( E )$ and $A ( E ^ { * } )$. For $n > 1$ there is a converse: If $E$ is holomorphically convex and if $\mu \mapsto \tilde{\mu}$ is bijective, then $E$ must in fact be $\mathbf{C}$-convex. Cf. Duality in complex analysis.

The surjectivity of the Fantappiè transform is closely related to integral representation formulas for holomorphic functions (cf. also Integral representations in multi-dimensional complex analysis). Let $f$ be holomorphic in a neighbourhood of a $\mathbf{C}$-convex compact set $E$, and let $\Omega$ be a smoothly bounded small open neighbourhood of $E$. By the Michael selection theorem one may choose, for every $\zeta \in \partial \Omega$, a hyperplane $s ( \zeta ) \in E ^ { * }$ depending in a smooth way on $\zeta$. Letting $s$ also denote the differential form $s _ { 1 } ( \zeta ) d \zeta _ { 1 } + \ldots + s _ { n } ( \zeta ) d \zeta _ { n }$, one then has the Cauchy–Fantappiè formula

\begin{equation*} f ( z ) = \frac { 1 } { ( 2 \pi i ) ^ { n } } \int _ { \partial \Omega } \frac { f ( \zeta ) s \wedge ( \overline { \partial } s ) ^ { n - 1 } } { \langle \zeta - z , s \rangle ^ { n } } ,\; z \in E. \end{equation*}

Here, the integral kernel is homogeneous in $s$, so one can replace $s$ by the new section $\sigma = - s / \langle s , \zeta \rangle$ and then obtain

\begin{equation*} f ( z ) = \frac { 1 } { ( 2 \pi i ) ^ { n } } \int _ { \partial \Omega } \frac { f ( \zeta ) \sigma \wedge ( \overline { \partial } \sigma ) ^ { n - 1 } } { ( 1 + \langle z , \sigma \rangle ) ^ { n } } ,\, z \in E. \end{equation*}

This integral formula has the following discrete analogue: Any function $f$, holomorphic in a neighbourhood of a $\mathbf{C}$-convex compact set $E \ni 0$, has a decomposition into partial fractions

\begin{equation*} f ( z ) = \sum _ { k = 1 } ^ { \infty } \frac { c _ { k } } { ( 1 + \langle z , \alpha _ { k } \rangle ) ^ { n } }, \end{equation*}

where all $a_k$ are contained in some compact subset of $E ^ { * }$, and $\sum | c_k| < \infty$. Hence the series converges uniformly in a neighbourhood of $E$.

If $E$ is only assumed to be weakly linearly convex, then the partial fraction representation becomes

\begin{equation*} f ( z ) = \sum _ { k = 1 } ^ { \infty } \frac { c _ { k } } { ( 1 + \langle z , a _ { k 1 } \rangle ) \ldots ( 1 + \langle z , a _ { k n } \rangle ) }, \end{equation*}

again with $\sum | c_k|< \infty$ and uniform convergence in a neighbourhood of $E$. Conversely, any holomorphically convex compact $E$ admitting such representations must necessarily be weakly linearly convex.

Existence of primitive functions.

For any vector $a \in \mathbf{C} ^ { n } \backslash \{ 0 \}$, let $( a , \partial )$ denote the first-order differential operator

\begin{equation*} f \mapsto \sum _ { k = 1 } ^ { n } a _ { k } \frac { \partial f } { \partial z _ { k } }. \end{equation*}

This directional derivative can, of course, be viewed as a mapping $A ( E ) \rightarrow A ( E )$, where $E$ is any domain in $\mathbf{C} ^ { n }$. If $E$ is $\mathbf{C}$-convex, then this mapping is surjective for any $a$, and conversely, if $E$ is bounded and the mapping $f \mapsto \langle a , \partial \rangle f$ is surjective for each $a$, then $E$ must be $\mathbf{C}$-convex. More generally, consider any linear differential operator $P ( \partial ) = P ( \partial / \partial z _ { 1 } , \dots , \partial / \partial z _ { n } )$ with constant coefficients, not all equal to zero, and regard it as a mapping $A ( E ) \rightarrow A ( E )$. Then, if $E$ is $\mathbf{C}$-convex, this mapping is surjective.

$\mathbf{C}$-convexity in other contexts.

Sets that are $\mathbf{C}$-convex occur naturally in several other connections, such as in complex polynomial approximation (in particular, in so-called Kergin interpolation), invariant metrics and pluri-potential theory, for instance in the work of L. Lempert. The fact that the dual complement of a convex set in $\mathbf{C} ^ { n }$ is not necessarily convex, but only $\mathbf{C}$-convex, is also a motivation for considering $\mathbf{C}$-convex sets.

Terminology.

There is a certain confusion in the literature as to the terminology connected with $\mathbf{C}$-convexity and the related convexity notions. The first ones to study weak linear convexity were H. Behnke and E. Peschl [a4], who used the term Planarkonvexität. Later, A. Martineau [a8] introduced the term convexité linéelle for what is here called linear convexity. At about the same time, L. Aizenberg [a2] and his school in Krasnoyarsk took up the Russian expression lineinaya vypuklost', which in English translations became "linear convexity" , to denote what is here called weak linear convexity. Some authors prefer to use the English adjective "lineal" rather than linear in this context, stressing the fact that the (original) French term is "linéel" and not "linéaire" . Until the 1990s, strong linear convexity was the term used for $\mathbf{C}$-convexity, whereas projective complement and conjugate set were used as synonyms for the dual complement $E ^ { * }$. In the above, the terminology adopted by L.V. Hörmander [a6] is followed.

References

[a1]	L. Aizenberg, "Decomposition of holomorphic functions of several complex variables into partial fractions" Sib. Math. J. , 8 (1967) pp. 859–872
[a2]	L. Aizenberg, "Linear convexity in $\mathbf{C} ^ { n }$ and the separation of singularities of holomorphic functions" Bull. Acad. Polon. Sci. Ser. Math. Astr. Phys. , 15 (1967) pp. 487–495 (In Russian) (French summary)
[a3]	M. Andersson, M. Passare, R. Sigurdsson, "Complex convexity and analytic functionals I" Science Inst. Univ. Iceland Preprint , June (1995)
[a4]	H. Behnke, E. Peschl, "Zur Theorie der Funktionen mehrerer komplexer Veränderlichen. Konvexität in bezug auf analytische Ebenen im kleinen und grossen" Math. Ann. , 111 (1935) pp. 158–177 MR1512986 Zbl 0011.16903 Zbl 61.0373.01
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[a6]	L. Hörmander, "Notions of convexity" , Progr. Math. , 127 , Birkhäuser (1994) MR1301332 Zbl 0835.32001
[a7]	C.O. Kiselman, "A differential inequality characterizing weak lineal convexity" Math. Ann. , 311 (1998) pp. 1–10 MR1624326 Zbl 0911.32031
[a8]	A. Martineau, "Sur la topologie des espaces de fonctions holomorphes" Math. Ann. , 163 (1966) pp. 62–88 MR0190697 Zbl 0138.38101
[a9]	A. Martineau, "Sur la notion d'ensemble fortement linéellement convexe" An. Acad. Brasil. Ci. , 40 (1968) pp. 427–435 MR0282292
[a10]	S.V. Znamenskii, "A geometric criterion for strong linear convexity" Funct. Anal. Appl. , 13 (1979) pp. 224–225 MR545375
[a11]	S.V. Znamenskii, L.N. Znamenskaya, "Spiral connectedness of the sections and projections of $\mathbf{C}$-convex sets" Math. Notes , 59 (1996) pp. 253–260