What are the names of the convex polyhedron with five, six, or seven vertices, where all vertices lie at distinct corners of a cube? I'm particularly interested in the five vertex case.
With five vertices, depending on which ones you choose, you can get an oblique square pyramid, an oblique rectangular pyramid, or an irregular triangular bipyramid with $S_3$ symmetry.
With six vertices, you can get a right isosceles triangular prism, an equilateral triangular right antiprism, or an irregular octahedron with $S_2$ symmetry.
I don't think the seven-vertex figure has any particular name.
Neither of these names are precise enough to express the fact that the vertices are chosen among the vertices of a cube.