Let $\Omega$ be a bounded convex domain in $\mathbb{R}^n$ and $u$ be a continuous convex function on the closure $\overline{\Omega}$. Given a boundary point $p\in\partial\Omega$, $u$ is said to have *infinite slop* at $p$ if $$\lim_{t\rightarrow0^+}\frac{u((1-t)p+tq)-u(p)}{t}=-\infty$$ for some $q\in\Omega$ (which implies the same limit for every $q\in\Omega$). Otherwise, $u$ is said to have *finite slop* at $p$.

On the other hand, the *Legrendre transform domain* of $u$ is the subset of the dual vector space $\mathbb{R}^{*n}$ where the Legendre transform of $u$ is defined (i.e. takes finite values). If $u$ is $C^1$ in $\Omega$, this domain is nothing but $\{Du(q)\mid q\in\Omega\}$.

The following question has an obvious positive answer when $n=1$, but I had some trouble trying to prove the "if" part in general:

Question.Is it true that for any convex function $u\in C^0(\overline{\Omega})$ with $u|_{\partial\Omega}=0$, the Legendre transform domain of $u$ is bounded if and only if $u$ has finite slope at every boundary point?

Is there a literature addressing this issue?