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 of some vector space is if it satisfies any of the following equivalent conditions: If is real and then If is real and with then Throughout, will be a map valued in the extended real numbers with a domain that is a convex subset of some vector space. The map is a if holds for any real and any with If this remains true of when the defining inequality () is replaced by the strict inequality then is called . Convex functions are related to convex sets. Specifically, the function is convex if and only if its is a convex set. The epigraphs of extended real-valued functions play a role in convex analysis that is analogous to the role played by graphs of real-valued function in real analysis. Specifically, the epigraph of an extended real-valued function provides geometric intuition that can be used to help formula or prove conjectures. The domain of a function is denoted by while its is the set The function is called if and for Alternatively, this means that there exists some in the domain of at which and is also equal to In words, a function is if its domain is not empty, it never takes on the value and it also is not identically equal to If is a proper convex function then there exist some vector and some such that for every where denotes the dot product of these vectors. Convex conjugate The of an extended real-valued function (not necessarily convex) is the function from the (continuous) dual space of and where the brackets denote the canonical duality The of is the map defined by for every If denotes the set of -valued functions on then the map defined by is called the . Subdifferential set and the Fenchel-Young inequality If and then the is For example, in the important special case where is a norm on , it can be shown that if then this definition reduces down to: and For any and which is called the . This inequality is an equality (i.e. ) if and only if It is in this way that the subdifferential set is directly related to the convex conjugate Biconjugate The of