Editing Convex conjugate (section)

== Examples ==
For more examples, see {{Section link||Table of selected convex conjugates}}.
* The convex conjugate of an [[affine function]] <math> f(x) = \left\langle a, x \right\rangle - b</math> is <math display="block"> f^{*}\left(x^{*} \right)
= \begin{cases} b,      & x^{*}   =  a
             \\ +\infty, & x^{*}  \ne a.
  \end{cases}
</math>
* The convex conjugate of a [[power function]] <math> f(x) = \frac{1}{p}|x|^p, 1 < p < \infty </math> is <math display="block">
f^{*}\left(x^{*} \right) = \frac{1}{q}|x^{*}|^q, 1<q<\infty, \text{where} \tfrac{1}{p} + \tfrac{1}{q} = 1.</math>
* The convex conjugate of the [[absolute value]] function <math>f(x) = \left| x \right|</math> is <math display="block">
f^{*}\left(x^{*} \right)
= \begin{cases} 0,      & \left|x^{*} \right| \le 1
             \\ \infty, & \left|x^{*} \right|  >  1.
  \end{cases}
</math>
* The convex conjugate of the [[exponential function]] <math>f(x)= e^x</math> is <math display="block">
f^{*}\left(x^{*} \right)
= \begin{cases} x^{*} \ln x^{*} - x^{*}      , & x^{*}  > 0
             \\ 0                            , & x^{*}  = 0
             \\ \infty                       , & x^{*}  < 0.
  \end{cases}
</math>

The convex conjugate and Legendre transform of the exponential function agree except that the [[domain of a function|domain]] of the convex conjugate is strictly larger as the Legendre transform is only defined for [[positive real numbers]].

===Connection with expected shortfall (average value at risk)===

See [https://link.springer.com/article/10.1007/s10107-014-0801-1 this article for example.]

Let ''F'' denote a [[cumulative distribution function]] of a [[random variable]]&nbsp;''X''. Then ([[Integration by parts|integrating by parts]]),
<math display="block">f(x):= \int_{-\infty}^x F(u) \, du = \operatorname{E}\left[\max(0,x-X)\right] = x-\operatorname{E} \left[\min(x,X)\right]</math>
has the convex conjugate
<math display="block">f^{*}(p)= \int_0^p F^{-1}(q) \, dq = (p-1)F^{-1}(p)+\operatorname{E}\left[\min(F^{-1}(p),X)\right] 
 = p F^{-1}(p)-\operatorname{E}\left[\max(0,F^{-1}(p)-X)\right].</math>

=== Ordering ===
A particular interpretation has the transform
<math display="block">f^\text{inc}(x):= \arg \sup_t t\cdot x-\int_0^1 \max\{t-f(u),0\} \, du,</math>
as this is a nondecreasing rearrangement of the initial function ''f''; in particular, <math>f^\text{inc}= f</math> for ''f'' nondecreasing.