Complex convexity is a general term in
complex geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...
.
Definition
A set
in
is called if its intersection with any complex line is contractible.
Background
In
complex geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...
and
analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
, the notion of
convexity
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
and its generalizations play an important role in understanding
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
behavior. Examples of classes of functions with a rich structure are, in addition to the convex functions, the
subharmonic function
In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory.
Intuitively, subharmonic functions are related to convex functio ...
s and the
plurisubharmonic functions.
Geometrically, these classes of functions correspond to convex domains and pseudoconvex domains, but there are also other types of domains, for instance lineally convex domains which can be generalized using
convex analysis
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.
Convex sets
A subset C \subseteq X of som ...
.
A great deal is already known about these domains, but there remain some fascinating, unsolved problems. This theme is mainly theoretical, but there are computational aspects of the domains studied, and these computational aspects are certainly worthy of further study.
References
Complex analysis
Convex analysis
External links
{{geometry-stub