Essential smoothness of a convex function

An extended-real convex function $\Lambda :{\mathbb{R}}^d\to \mathbb{R}\cup \{+\infty \}$ is essentially smooth if its effective domain has nonempty interior, it is differentiable throughout that interior, and $\Vert \nabla \Lambda ({\theta }_n)\Vert \to \infty$ whenever interior points converge to a finite boundary point of the domain. A differentiable convex function finite on all of ${\mathbb{R}}^d$ satisfies the boundary condition vacuously. The cumulant-generating function $\log (r/(r-\theta ))$ on $\theta <r$ has derivative $(r-\theta {)}^{-1}$ and is essentially smooth. Restricting its domain to $\theta <\lambda <r$ destroys this property. This distinction matters in the lower bounds of the Gärtner–Ellis theorem.