The domain of lsc convex functions How to see that $int(dom(f)) \neq \emptyset$ where $f$ is a proper convex lower semi-continuous function from a Banach space to $\mathbb{R} \cup \{+\infty\}.$
A function $$f(x)=\begin{cases}+\infty&amp;\text{for }x\ne 0\\0&amp;\text{for }x=0\end{cases}$$ is proper convex lsc function while its domain is a singleton $\{0\}.$
A function $$f(x)=\begin{cases}+\infty&amp;\text{for }x\ne 0\\0&amp;\text{for }x=0\end{cases}$$ is proper convex lsc function while its domain is a singleton $\{0\}.$