On Maximal Domains for C-Convex Functions and Convex Extensions

Let $f$ be a real valued function with the domain $\dom(f)$ in some vector space $X$ and let $\mathfrak{C}$ be the collection of convex subsets of $X$. The following two questions are investigated; 1. Do there exist maximal convex restrictions $g$ of $f$ with $\dom(g) \in \mathfrak{C}$? 2. If $f$ is convex with $\dom(f)\in \mathfrak{C}$, do there exist maximal convex extension $g$ of $f$ with $\dom(g)\in \mathfrak{C}$? We will show that the answer to both questions is positive under a certain condition on $\mathfrak{C}$.