Existence of a convex extension of a preference relation

In this paper we say that a preference (i.e. irreflexive) relationP isregular (or aweak order) if both it and its non-comparability relation are transitive; we also say that a preference relationP ^* is aconvex extension of another preference relationP ifP?P ^* holds andP ^* is regular and convex-valued. We prove that a convex extension ofP exists if and only if every non-empty and finite set of alternativesA is not included in the convex hull of ∪ _x∈A P(x).