In order to guarantee the existence of the Legendre transform of a function $f:\mathbb{R}\rightarrow\mathbb{R}$, one usually has to know that $f$ is convex.
When performing a Legendre transformation on $U$ with respect to $S$ (obtaining the Helmholtz free energy) one is seemingly assuming that $U_{V,N}(S)$ is a convex function. Is this true? Why?
I know that $\frac{\partial U}{\partial S}=T$, so that $U$ is convex if $\frac{\partial T}{\partial S}>0$ (for all $S,V,N$). This certainly seems to be the case.
