Are Legendre transforms of non-convex functions useful?

Question

1. Do Legendre transforms have any applications that do not appeal to convexity?

2. What is the intuitive interpretation of the Legendre transform of a non-convex function?

Answer 1

I can answer your second question.

Recall that by convex duality, applying the Legendre transform twice on a convex (and coercive) function gives you back the original function. Now even if you plug in a non-convex function $W$ instead, the Legendre transform $W^*$ will be convex and coercive (one can see that by the definition) and so is $V:=(W^*)^*$. So clearly the convex duality will not carry over to the non-convex case. However one can show (under appropriate assumptions of course) that $V$ is the convexification or convex hull of $W$, that is roughly (here $W:\mathbb{R}\rightarrow\mathbb{R}$) the function whose graph is the boundary curve of the convex hull of the set $\{(x,y)\in\mathbb{R}^2\,|\,y\ge f(x)\}$. By convex duality we conclude that $V^*=((W^*)^*)^*=W^*$, so the Legendre transform of a non-convex function is the Legendre transform of it's convexification.