Editing Convex function (section)

=== Functions of several variables ===

* A function that is marginally convex in each individual variable is not necessarily (jointly) convex. For example, the function <math>f(x, y) = x y</math> is [[bilinear map|marginally linear]], and thus marginally convex, in each variable, but not (jointly) convex.
* A function <math>f : X \to [-\infty, \infty]</math> valued in the [[extended real numbers]] <math>[-\infty, \infty] = \R \cup \{\pm\infty\}</math> is convex if and only if its [[Epigraph (mathematics)|epigraph]] <math display=block>\{(x, r) \in X \times \R ~:~ r \geq f(x)\}</math> is a convex set.
* A differentiable function <math>f</math> defined on a convex domain is convex if and only if <math>f(x) \geq f(y) + \nabla f(y)^T \cdot (x-y)</math> holds for all <math>x, y</math> in the domain.
* A twice differentiable function of several variables is convex on a convex set if and only if its [[Hessian matrix]] of second [[partial derivative]]s is [[Positive-definite matrix|positive semidefinite]] on the interior of the convex set.
* For a convex function <math>f,</math> the [[sublevel set]]s <math>\{x : f(x) < a\}</math> and <math>\{x : f(x) \leq a\}</math> with <math>a \in \R</math> are convex sets. A function that satisfies this property is called a '''{{em|[[quasiconvex function]]}}''' and may fail to be a convex function.
* Consequently, the set of [[Arg min|global minimisers]] of a convex function <math>f</math> is a convex set: <math>{\operatorname{argmin}}\,f</math> - convex.
* Any [[local minimum]] of a convex function is also a [[global minimum]].  A {{em|strictly}} convex function will have at most one global minimum.<ref>{{cite web | url=https://math.stackexchange.com/q/337090 | title=If f is strictly convex in a convex set, show it has no more than 1 minimum | publisher=Math StackExchange | date=21 Mar 2013 | access-date=14 May 2016}}</ref>
* [[Jensen's inequality]] applies to every convex function <math>f</math>. If <math>X</math> is a random variable taking values in the domain of <math>f,</math> then <math>\operatorname{E}(f(X)) \geq f(\operatorname{E}(X)),</math> where <math>\operatorname{E}</math> denotes the [[Expected value|mathematical expectation]]. Indeed, convex functions are exactly those that satisfies the hypothesis of [[Jensen's inequality]]. 
* A first-order [[homogeneous function]] of two positive variables <math>x</math> and <math>y,</math> (that is, a function satisfying <math>f(a x, a y) = a f(x, y)</math> for all positive real <math>a, x, y > 0</math>) that is convex in one variable must be convex in the other variable.<ref>Altenberg, L., 2012. Resolvent positive linear operators exhibit the reduction phenomenon. Proceedings of the National Academy of Sciences, 109(10), pp.3705-3710.</ref>