I have a quick question about the legendre transformation. Assume we have a function given: $f(x,y,z)$ which is convex in all three variables. I want to take minus the legendre transformation of this function, where I'm sending $x \rightarrow p$.
This means: $f^*(p,y,z) = - sup_x ( xp - f ) = inf_x (f - xp) $
My question now is, in which variables is the legendre transformation convex and in which concave?
My reasoning would be: No matter the function one starts with, the legendre transform is convex in the transformed variable. This means my function is concave in $p$ since we have a minus sign. The variables which don't get transformed keep their properties, so $sup_x(xp- f)$ should still be convex in $y,z$. So the minus sign makes them concave, right? So in the end my function should be concave in all three variables.
This question arose from a phyiscs problem, where I'm trying to get the free energy $F(T,V,N)$ which is minus the legendre transformation of the inner energy $U(S,V,N)$. With my reasoning $F$ should be concave in all 3 variables, but my textbook says it is concave in $T$ and convex in $V$ and $N$.
Cheers
Almost correct - up to the minus sign. The expression $xp-f(x,y,z)$ is concave in $y$ and $z$, due to the minus sign in front of $f$. Taking the supremum over $x$ maintains this property. Hence, the transform is concave in $y,z$ and negative transform is convex in $y,z$. As your book said.