Editing Legendre transformation (section)

==Legendre transformation in more than one dimension==
For a differentiable real-valued function on an [[open set|open]] convex subset {{mvar|U}} of {{math|'''R'''<sup>''n''</sup>}} the Legendre conjugate of the pair {{math|(''U'', ''f'')}} is defined to be the pair {{math|(''V'', ''g'')}}, where {{mvar|V}} is the image of {{mvar|U}} under the [[gradient]] mapping {{math|''Df''}}, and {{mvar|g}} is the function on {{mvar|V}} given by the formula
<math display="block">g(y) = \left\langle y, x \right\rangle - f(x), \qquad x = \left(Df\right)^{-1}(y)</math>
where
<math display="block">\left\langle u,v\right\rangle = \sum_{k=1}^n u_k \cdot v_k</math>

is the [[scalar product]] on {{math|'''R'''<sup>''n''</sup>}}. The multidimensional transform 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=http://maze5.net/?page_id=733 |title=Legendre Transform {{pipe}} Nick Alger // Maps, art, etc |access-date=2011-01-26 |archive-url=https://web.archive.org/web/20150312152731/http://maze5.net/?page_id=733 |archive-date=2015-03-12 |url-status=dead }}</ref> This can be seen as consequence of the following two observations. On the one hand, the hyperplane tangent to the epigraph of <math>f</math> at some point <math>(\mathbf x, f(\mathbf x))\in U\times \mathbb{R}</math> has normal vector <math>(\nabla f(\mathbf x),-1)\in\mathbb{R}^{n+1}</math>. On the other hand, any closed convex set <math>C\in\mathbb{R}^m</math> can be characterized via the set of its [[Supporting hyperplane|supporting hyperplanes]] by the equations <math>\mathbf x\cdot\mathbf n = h_C(\mathbf n)</math>, where <math>h_C(\mathbf n)</math> is the [[support function]] of <math>C</math>. But the definition of Legendre transform via the maximization matches precisely that of the support function, that is, <math>f^*(\mathbf x)=h_{\operatorname{epi}(f)}(\mathbf x,-1) </math>. We thus conclude that the Legendre transform characterizes the epigraph in the sense that the tangent plane to the epigraph at any point <math>(\mathbf x,f(\mathbf x))</math>  is given explicitly by<math display="block">\{\mathbf z\in\mathbb{R}^{n+1}: \,\, \mathbf z\cdot \mathbf x= f^*(\mathbf x)\}. </math>

Alternatively, if {{mvar|X}} is a [[vector space]] and {{math|''Y''}} is its [[dual space|dual vector space]], then for each point {{mvar|x}} of {{math|''X''}} and {{math|''y''}} of {{math|''Y''}}, there is a natural identification of the [[cotangent space]]s {{math|T*''X<sub>x</sub>''}} with {{math|''Y''}} and {{math|T*''Y<sub>y</sub>''}} with {{math|''X''}}. If {{mvar|f}} is a real differentiable function over {{math|''X''}}, then its [[exterior derivative]], {{math|''df''}}, is a section of the [[cotangent bundle]] {{math|T*''X''}} and as such, we can construct a map from {{math|''X''}} to {{math|''Y''}}. Similarly, if {{mvar|g}} is a real differentiable function over {{math|''Y''}}, then {{math|''dg''}} defines a map from {{math|''Y''}} to {{math|''X''}}. If both maps happen to be inverses of each other, we say we have a Legendre transform. The notion of the [[tautological one-form]] is commonly used in this setting.

When the function is not differentiable, the Legendre transform can still be extended, and is known as the [[Legendre-Fenchel transformation]]. In this more general setting, a few properties are lost: for example, the Legendre transform is no longer its own inverse (unless there are extra assumptions, like [[convex function|convexity]]).