In equivalent but more mathematical terms, the convex hull of a set of points S (in some vector space over the real numbers or other field of characteristic 0) is the "smallest" convex set containing S. Such a set exists: the intersection of a family of convex sets is itself convex, so the convex hull of S is merely the intersection of all convex sets containing S.
Equivalently, since every convex set is the intersection of half spaces, the convex hull of S is the intersection of all half spaces containing S.