What is cube/hypercube/cross polytope?

A subset $P$ of $R^d$ is called a unit $d$-cube if it is the convex hull of all $2^d$ points with components 0 or $1$. It has exactly $2^d$ vertices and $2 d$ facets. A cube or hypercube is a convex polytope which is isomorphic to the $d$-cube for some $d$.

A dual (see 2.4) of a cube is called a cross polytope.