Need help understanding the scaling properties of the Legendre transformationAt the wikipedia page for the Legendre transformation, there is a section on scaling properties where it saysf(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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Need help understanding the scaling properties of the Legendre transformationAt the wikipedia page for the Legendre transformation, there is a section on scaling properties where it saysf(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:&quot;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&#039;t see how the scaling properties hold. I&#039;d appreciate if someone could spell this out for slow me.2) I don&#039;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.)

utripljigmp · Accepted Answer

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maskorfclp · Accepted Answer

Call your two formulas (1) and (2). Suppose you have a homogeneous function f of degree r&amp;gt;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⋅(k1−rs)=k−rf⋅(s)  Now, if you call k1−r=λ,then k−r=λrr−1, that is  f⋅(λs)=λrr−1f⋅(s)  Threfore f* is homogeneous of degree q=rr−1. 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.

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

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