Need help understanding the scaling properties of the Legendre transformation At the wikipedia p

Jennifer Beasley 2022-02-15 Answered
Need help understanding the scaling properties of the Legendre transformation
At the wikipedia page for the Legendre transformation, there is a section on scaling properties where it says
f(x)=ag(x)f(p)=ag(pa)
and f(x)=g(ax)f(p)=g(pa)
where f*(p) and g*(p) are the Legendre transformations of f(x) and g(x), respectively, and a is a scale factor.
Also, it says:
"It follows that if a function is homogeneous of degree r then its image under the Legendre transformation is a homogeneous function of degree s, where 1r+1s=1.
1) I don't see how the scaling properties hold. I'd appreciate if someone could spell this out for slow me.
2) I don't see how the relation between the degrees of homogeneity (r,s) follows from the scaling properties. Need some spelling out here too.
3) If the relation 1r+1s=1 is true, then could this be used to prove that linearly homogeneous functions are not convex/concave? (Because convex/concave functions have a Legendre transformation, and r=1 would imply s=, which is absqrt and thus tantamount to saying that a function with r=1 has no Legendre transformation. No Legendre transformation then implies no convexity/concavity.)
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Expert Answer

utripljigmp
Answered 2022-02-16 Author has 12 answers
maskorfclp
Answered 2022-02-17 Author has 12 answers
Call your two formulas (1) and (2). Suppose you have a homogeneous function f of degree r>1 (this is the range where it is convex). That means f(kx)=krf(x).
Now take the transform of both sides at p. Use (2) for the left hand side and (1) for the right hand side.
f(pk)=krf(pkr).
Now, if you call pk=s the above relation becomes
f(k1rs)=krf(s)
Now, if you call k1r=λ,then kr=λrr1, that is
f(λs)=λrr1f(s)
Threfore f* is homogeneous of degree q=rr1. It is straightforward to check that 1r+1q=1. When r=1 your transform is zero at p= slope of the line and is infinity elsewhere.You can interpret this formally as being homogeneous of infinite degree (both zero and infinity satisfy such homogeneity condition formally). A function like this is convex.
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