Hidden vertices in extensions of polytopes

Some widely known compact extended formulations have the property that each vertex of the corresponding extension polytope is projected onto a vertex of the target polytope. In this paper, we prove that for heptagons with vertices in general position none of the minimum size extensions has this property. Additionally, for any d≥2 we construct a family of d-polytopes such that at least 19 of all vertices of any of their minimum size extensions is not projected onto vertices.