Domains of convexity (Q1842058)

Let \(D\subset {\mathbf R}^n\) be a closed convex set with nonempty interior. The authors prove that there exists a convex locally lipschitzian function \(f:D \to {\mathbf R}\) which does not admit any convex finite-valued extension outside \(D\) if and only if the following condition is satisfied: \(D= {\mathbf R}^n\) or through every point of the boundary of \(D\) there passes at least one half-line lying on the boundary of \(D\).