The legendre transform reaches the supremum it's defined with.

As is known, the Legendre transform g of a convex function f is defined as :

$g(p)=\sup_{x \in \mathbb{R}^n} p(x)-f(x)$With $p\in\mathcal{L}(\mathbb{R}^n,\mathbb{R})$.

However, it can be shown that the maximizer of the expression above exists and here lies my question.

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a convex smooth function satisfying :

$\lim_{\| x\| \to \infty} \frac{f(x)}{\|x\|} = \infty.$

If $p\in\mathcal{L}(\mathbb{R}^n,\mathbb{R})$, can you show that the superlevel sets :{q, $p(q)-f(q)\geq c$} are compact ?

This would help me show that a supremum exists and is reached.