Editing Convex function (section)

==Operations that preserve convexity==

* <math>-f</math> is concave if and only if <math>f</math> is convex.
* If <math>r</math> is any real number then <math>r + f</math> is convex if and only if <math>f</math> is convex.
* Nonnegative weighted sums:
**if <math>w_1, \ldots, w_n \geq 0</math> and <math>f_1, \ldots, f_n</math> are all convex, then so is <math>w_1 f_1 + \cdots + w_n f_n.</math> In particular, the sum of two convex functions is convex.
**this property extends to infinite sums, integrals and expected values as well (provided that they exist).
* Elementwise maximum: let <math>\{f_i\}_{i \in I}</math> be a collection of convex functions. Then <math>g(x) = \sup\nolimits_{i \in I} f_i(x)</math> is convex. The domain of <math>g(x)</math> is the collection of points where the expression is finite. Important special cases:
**If <math>f_1, \ldots, f_n</math> are convex functions then so is <math>g(x) = \max \left\{f_1(x), \ldots, f_n(x)\right\}.</math>
**[[Danskin's theorem]]: If <math>f(x,y)</math> is convex in <math>x</math> then <math>g(x) = \sup\nolimits_{y\in C} f(x,y)</math> is convex in <math>x</math> even if <math>C</math> is not a convex set.
* Composition:
**If <math>f</math> and <math>g</math> are convex functions and <math>g</math> is non-decreasing over a univariate domain, then <math>h(x) = g(f(x))</math> is convex. For example, if <math>f</math> is convex, then so is <math>e^{f(x)}</math> because <math>e^x</math> is convex and monotonically increasing.
**If <math>f</math> is concave and <math>g</math> is convex and non-increasing over a univariate domain, then <math>h(x) = g(f(x))</math> is convex.
**Convexity is invariant under affine maps: that is, if <math>f</math> is convex with domain <math>D_f \subseteq \mathbf{R}^m</math>, then so is <math>g(x) = f(Ax+b)</math>, where <math>A \in \mathbf{R}^{m \times n}, b \in \mathbf{R}^m</math> with domain <math>D_g \subseteq \mathbf{R}^n.</math>
* Minimization: If <math>f(x,y)</math> is convex in <math>(x,y)</math> then <math>g(x) = \inf\nolimits_{y\in C} f(x,y)</math> is convex in <math>x,</math> provided that <math>C</math> is a convex set and that <math>g(x) \neq -\infty.</math>
* If <math>f</math> is convex, then its perspective <math>g(x, t) = t f \left(\tfrac{x}{t} \right)</math> with domain <math>\left\{(x,t) : \tfrac{x}{t} \in \operatorname{Dom}(f), t > 0 \right\}</math> is convex.
* Let <math>X</math> be a vector space. <math>f : X \to \mathbf{R}</math> is convex and satisfies <math>f(0) \leq 0</math> if and only if <math>f(ax+by) \leq a f(x) + bf(y)</math> for any <math>x, y \in X</math> and any non-negative real numbers <math>a, b</math> that satisfy <math>a + b \leq 1.</math>