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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.}
Oka's proof of Levi's problem was that when the unramified Riemann domain over [54] was a domain of holomorphy (holomorphically convex), it was proved that it was necessary and sufficient that each boundary point of the domain of holomorphy is an Oka pseudoconvex.
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 .
In complex geometry and analysis, the notion of convexity and its generalizations play an important role in understanding function behavior. Examples of classes of functions with a rich structure are, in addition to the convex functions, the subharmonic functions and the plurisubharmonic functions.
This definition is valid for any function, ... A closed proper convex function f is the pointwise supremum of the collection of all affine functions h such that h ≤ ...
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.
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 ...
The term domain is also commonly used in a different sense in mathematical analysis: a domain is a non-empty connected open set in a topological space. In particular, in real and complex analysis , a domain is a non-empty connected open subset of the real coordinate space R n {\displaystyle \mathbb {R} ^{n}} or the complex coordinate space C n ...