Editing Convex conjugate (section)

== Definition ==

Let <math>X</math> be a [[real number|real]] [[topological vector space]] and let <math>X^{*}</math> be the [[dual space]] to <math>X</math>. Denote by 

:<math>\langle \cdot , \cdot \rangle : X^{*} \times X \to \mathbb{R}</math>

the canonical [[dual pair]]ing, which is defined by <math>\left\langle x^*, x \right\rangle \mapsto x^* (x).</math> 

For a function <math>f : X \to \mathbb{R} \cup \{ - \infty, + \infty \}</math> taking values on the [[extended real number line]], its '''{{em|convex conjugate}}''' is the function

:<math>f^{*} : X^{*} \to \mathbb{R} \cup \{ - \infty, + \infty \}</math>

whose value at <math>x^* \in X^{*}</math> is defined to be the [[supremum]]:

:<math>f^{*} \left( x^{*} \right) := \sup \left\{ \left\langle x^{*}, x \right\rangle - f (x) ~\colon~ x \in X \right\},</math>

or, equivalently, in terms of the [[infimum]]:

:<math>f^{*} \left( x^{*} \right) := - \inf \left\{ f (x) - \left\langle x^{*}, x \right\rangle ~\colon~ x \in X \right\}.</math>

This definition can be interpreted as an encoding of the [[convex hull]] of the function's [[Epigraph (mathematics)|epigraph]] in terms of its [[supporting hyperplane]]s.<ref>{{cite web|url=https://physics.stackexchange.com/a/9360/821 |title=Legendre Transform |accessdate=April 14, 2019}}</ref>