On Maximal Domains for C-Convex Functions and Convex Extensions

Let

be a real valued function with the domain

\operatorname{dom}(f) dom ⁡ ( f )

in some vector space

X X

and let

\mathfrak{C} C

be the collection of convex subsets of

X X

. The following two questions are investigated; 1. Do there exist maximal convex restrictions

g g

of

f f

with

\operatorname{dom}(g) \in \mathfrak{C} dom ⁡ ( g ) ∈ C

? 2. If

f f

is convex with

\operatorname{dom}(f)\in \mathfrak{C} dom ⁡ ( f ) ∈ C

, do there exist maximal convex extension

g g

of

f f

with

\operatorname{dom}(g)\in \mathfrak{C} dom ⁡ ( g ) ∈ C

? We will show that the answer to both questions is positive under a certain condition on

\mathfrak{C} C

.