Editing Convex function (section)

==Definition==
[[File:Convex 01.ogg|thumb|right|Visualizing a convex function and Jensen's Inequality]]

Let <math>X</math> be a [[Convex set|convex subset]] of a real [[vector space]] and let <math>f: X \to \R</math> be a function.

Then <math>f</math> is called '''{{em|convex}}''' if and only if any of the following equivalent conditions hold:

<ol start=1>
<li>For all <math>0 \leq t \leq 1</math> and all <math>x_1, x_2 \in X</math>:
<math display=block>f\left(t x_1 + (1-t) x_2\right) \leq t f\left(x_1\right) + (1-t) f\left(x_2\right)</math>
The right hand side represents the straight line between <math>\left(x_1, f\left(x_1\right)\right)</math> and <math>\left(x_2, f\left(x_2\right)\right)</math> in the graph of <math>f</math> as a function of <math>t;</math> increasing <math>t</math> from <math>0</math> to <math>1</math> or decreasing <math>t</math> from <math>1</math> to <math>0</math> sweeps this line. Similarly, the argument of the function <math>f</math> in the left hand side represents the straight line between <math>x_1</math> and <math>x_2</math> in <math>X</math> or the <math>x</math>-axis of the graph of <math>f.</math> So, this condition requires that the straight line between any pair of points on the curve of <math>f</math> be above or just meeting the graph.<ref>{{Cite web|last=|first=|date=|title=Concave Upward and Downward|url=https://www.mathsisfun.com/calculus/concave-up-down-convex.html|url-status=live|archive-url=https://web.archive.org/web/20131218034748/http://www.mathsisfun.com:80/calculus/concave-up-down-convex.html |archive-date=2013-12-18 |access-date=|website=}}</ref>
</li>
<li>For all <math>0 < t < 1</math> and all <math>x_1, x_2 \in X</math> such that <math>x_1\neq x_2</math>:
<math display=block>f\left(t x_1 + (1-t) x_2\right) \leq t f\left(x_1\right) + (1-t) f\left(x_2\right)</math>

The difference of this second condition with respect to the first condition above is that this condition does not include the intersection points (for example, <math>\left(x_1, f\left(x_1\right)\right)</math> and <math>\left(x_2, f\left(x_2\right)\right)</math>) between the straight line passing through a pair of points on the curve of <math>f</math> (the straight line is represented by the right hand side of this condition) and the curve of <math>f;</math> the first condition includes the intersection points as it becomes <math>f\left(x_1\right) \leq f\left(x_1\right)</math> or <math>f\left(x_2\right) \leq f\left(x_2\right)</math> at <math>t = 0</math> or <math>1,</math> or <math>x_1 = x_2.</math> In fact, the intersection points do not need to be considered in a condition of convex using <math display=block>f\left(t x_1 + (1-t) x_2\right) \leq t f\left(x_1\right) + (1-t) f\left(x_2\right)</math> because <math>f\left(x_1\right) \leq f\left(x_1\right)</math> and <math>f\left(x_2\right) \leq f\left(x_2\right)</math> are always true (so not useful to be a part of a condition). 
</li>
</ol>

The second statement characterizing convex functions that are valued in the real line <math>\R</math> is also the statement used to define '''{{em|convex functions}}''' that are valued in the [[extended real number line]] <math>[-\infty, \infty] = \R \cup \{\pm\infty\},</math> where such a function <math>f</math> is allowed to take <math>\pm\infty</math> as a value. The first statement is not used because it permits <math>t</math> to take <math>0</math> or <math>1</math> as a value, in which case, if <math>f\left(x_1\right) = \pm\infty</math> or <math>f\left(x_2\right) = \pm\infty,</math> respectively, then <math>t f\left(x_1\right) + (1 - t) f\left(x_2\right)</math> would be undefined (because the multiplications <math>0 \cdot \infty</math> and <math>0 \cdot (-\infty)</math> are undefined). The sum <math>-\infty + \infty</math> is also undefined so a convex extended real-valued function is typically only allowed to take exactly one of <math>-\infty</math> and <math>+\infty</math> as a value. 

The second statement can also be modified to get the definition of {{em|strict convexity}}, where the latter is obtained by replacing <math>\,\leq\,</math> with the strict inequality <math>\,<.</math> 
Explicitly, the map <math>f</math> is called '''{{em|strictly convex}}''' if and only if for all real <math>0 < t < 1</math> and all <math>x_1, x_2 \in X</math> such that <math>x_1 \neq x_2</math>:
<math display=block>f\left(t x_1 + (1-t) x_2\right) < t f\left(x_1\right) + (1-t) f\left(x_2\right)</math>

A strictly convex function <math>f</math> is a function that the straight line between any pair of points on the curve <math>f</math> is above the curve <math>f</math> except for the intersection points between the straight line and the curve.  An example of a function which is convex but not strictly convex is <math>f(x,y) = x^2 + y</math>. This function is not strictly convex because any two points sharing an x coordinate will have a straight line between them, while any two points NOT sharing an x coordinate will have a greater value of the function than the points between them.

The function <math>f</math> is said to be '''{{em|[[Concave function|concave]]}}''' (resp. '''{{em|strictly concave}}''') if <math>-f</math> (<math>f</math> multiplied by −1) is convex (resp. strictly convex).