At the wikipedia page for the Legendre transformation, there is a section on scaling properties where it says
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
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