Domains of Convexity

A closed convex set D ⊂ Rn, with nonempty interior, is said to be convexity domain if there exists a convex locally lipschitzian function g : D → R which does not admit any convex finite-valued extension outside D. It is shown that D is the convexity domain iff for each x ∈ bd(D) there exists a half-line passing through x and lying on the boundary of D. Other characterizations of convexity domains are given as well.