Editing Legendre transformation (section)

===Definition in one-dimensional real space===
Let <math>I \sub \R</math> be an [[Interval (mathematics)|interval]], and <math>f:I \to \R</math> a [[convex function]]; then the ''Legendre transform'' ''of'' <math>f</math> is the function <math>f^*:I^* \to \R</math> defined by
<math display="block">f^*(x^*) = \sup_{x\in I}(x^*x-f(x)),\ \ \ \ I^*= \left \{x^*\in \R:\sup_{x\in I}(x^*x-f(x)) <\infty \right \}</math>
where <math display="inline">\sup</math> denotes the [[Infimum and supremum|supremum]] over <math>I</math>, e.g., <math display="inline">x</math> in <math display="inline">I</math> is chosen such that <math display="inline">x^*x - f(x)</math> is maximized at each <math display="inline">x^*</math>, or <math display="inline">x^*</math> is such that <math>x^*x-f(x)</math> has a bounded value throughout <math display="inline">I</math> (e.g., when <math>f(x)</math> is a linear function).

The function <math>f^*</math> is called the [[convex conjugate]] function of <math>f</math>. For historical reasons (rooted in analytic mechanics), the conjugate variable is often denoted <math>p</math>, instead of <math>x^*</math>. If the convex function <math>f</math> is defined on the whole line and is everywhere [[Differentiable function|differentiable]], then
<math display="block">f^*(p)=\sup_{x\in I}(px - f(x)) = \left( p x - f(x) \right)|_{x = (f')^{-1}(p)} </math>
can be interpreted as the negative of the [[y-intercept|<math>y</math>-intercept]] of the [[tangent line]] to the [[Graph of a function|graph]] of <math>f</math> that has slope <math>p</math>.