Editing Legendre transformation (section)

==Further properties==

===Scaling properties===
The Legendre transformation has the following scaling properties: For {{math|''a'' > 0}},

<math display="block">f(x) = a \cdot g(x) \Rightarrow f^\star(p) = a \cdot g^\star\left(\frac{p}{a}\right) </math>
<math display="block">f(x) = g(a \cdot x) \Rightarrow f^\star(p) = g^\star\left(\frac{p}{a}\right).</math>

It follows that if a function is [[homogeneous function|homogeneous of degree {{mvar|r}}]] then its image under the Legendre transformation is a homogeneous function of degree {{mvar|s}}, where {{math|1=1/''r'' + 1/''s'' = 1}}. (Since {{math|1=''f''(''x'') = ''x<sup>r</sup>''/''r''}}, with {{math|''r'' > 1}}, implies {{math|1=''f''*(''p'') = ''p<sup>s</sup>''/''s''}}.) Thus, the only monomial whose degree is invariant under Legendre transform is the quadratic.

===Behavior under translation===
<math display="block"> f(x) = g(x) + b \Rightarrow f^\star(p) = g^\star(p) - b</math>
<math display="block"> f(x) = g(x + y) \Rightarrow f^\star(p) = g^\star(p) - p \cdot y </math>

===Behavior under inversion===
<math display="block"> f(x) = g^{-1}(x) \Rightarrow f^\star(p) = - p \cdot g^\star\left(\frac{1}{p} \right) </math>

===Behavior under linear transformations===
Let {{math|''A'' : '''R'''<sup>''n''</sup> → '''R'''<sup>''m''</sup>}} be a [[linear transformation]]. For any convex function {{mvar|f}} on {{math|'''R'''<sup>''n''</sup>}}, one has
<math display="block"> (A f)^\star = f^\star A^\star </math>
where {{math|''A''*}} is the [[adjoint operator]] of {{mvar|A}} defined by
<math display="block"> \left \langle Ax, y^\star \right \rangle = \left \langle x, A^\star y^\star \right \rangle, </math>
and {{math|''Af''}} is the ''push-forward'' of {{mvar|f}} along {{mvar|A}}
<math display="block"> (A f)(y) = \inf\{ f(x) : x \in X , A x = y \}. </math>

A closed convex function {{mvar|f}} is symmetric with respect to a given set {{mvar|G}} of [[orthogonal matrix|orthogonal linear transformations]],
<math display="block">f(A x) = f(x), \; \forall x, \; \forall A \in G </math>
[[if and only if]] {{math|''f''*}} is symmetric with respect to {{mvar|G}}.

===Infimal convolution===
The '''infimal convolution''' of two functions {{mvar|f}} and {{mvar|g}} is defined as

<math display="block"> \left(f \star_\inf g\right)(x) = \inf \left \{ f(x-y) + g(y) \, | \, y \in \mathbf{R}^n \right \}. </math>

Let {{math|''f''<sub>1</sub>, ..., ''f<sub>m</sub>''}} be proper convex functions on {{math|'''R'''<sup>''n''</sup>}}. Then

<math display="block"> \left( f_1 \star_\inf \cdots \star_\inf f_m \right)^\star = f_1^\star + \cdots + f_m^\star. </math>

===Fenchel's inequality===
For any function {{mvar|f}} and its convex conjugate {{math|''f'' *}} ''Fenchel's inequality'' (also known as the ''Fenchel–Young inequality'') holds for every {{math|''x'' ∈ ''X''}} and {{math|''p'' ∈ ''X''*}}, i.e., ''independent'' {{math|''x'', ''p''}} pairs,
<math display="block">\left\langle p,x \right\rangle \le f(x) + f^\star(p).</math>