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A third definition [14] for a strongly convex function, with parameter , is that, for all , in the domain and [,], (+ ()) + () ‖ ‖ Notice that this definition approaches the definition for strict convexity as m → 0 , {\displaystyle m\to 0,} and is identical to the definition of a convex function when m = 0. {\displaystyle m=0.}
The convex-hull operation is needed for the set of convex sets to form a lattice, in which the "join" operation is the convex hull of the union of two convex sets = = ( ()). The intersection of any collection of convex sets is itself convex, so the convex subsets of a (real or complex) vector space form a complete lattice .
The polynomially convex hull contains the holomorphically convex hull. The domain is called holomorphically convex if for every compact subset , ^ is also compact in G. Sometimes this is just abbreviated as holomorph-convex.
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. A great deal is already known about these domains, but there remain some fascinating, unsolved problems.
In convex analysis and variational analysis, a point (in the domain) at which some given function is minimized is typically sought, where is valued in the extended real number line [,] = {}. [1] Such a point, if it exists, is called a global minimum point of the function and its value at this point is called the global minimum (value) of the ...
Convex analysis includes not only the study of convex subsets of Euclidean spaces but also the study of convex functions on abstract spaces. 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.
However, there are pseudoconvex domains which are not geometrically convex. When G {\displaystyle G} has a C 2 {\displaystyle C^{2}} (twice continuously differentiable ) boundary , this notion is the same as Levi pseudoconvexity, which is easier to work with.
For a domain the following conditions are equivalent: . is a domain of holomorphy; is holomorphically convex; is pseudoconvex; is Levi convex - for every sequence of analytic compact surfaces such that , for some set we have (cannot be "touched from inside" by a sequence of analytic surfaces)