I am looking for the name (if there is one), simple properties and possible literature for the following class of polytopes (by polytope I mean the convex hull of finitely man points in $\Bbb R^n,n\ge 2$). For lack of another name, I call them transparent.
Definition. A polytope $P\subseteq \Bbb R^n$ is transparent if there exists a direction $v\in\Bbb R^n$, so that lines through the vertices of $P$ with direction $v$ do not intersect the interior of $P$.
Another characterization is, that the shadow (projection) of $P$ onto a hyperplane has all the vertices of $P$ on its boundary. I called them transparent because, if made from glass with opaque vertices, it is possible to look through their interior without any vertices in the way.
I wonder whether such polytopes were discussd in the literature before and whether there is a classification of these. I am especially interested in centrally symmetric polytopes with vertices on the unit sphere, but I think it is still interesting in the general case.
Examples. The following polytopes are transparent:
- Simplices, hypercubes and cross-polytopes of dimensions $\ge 2$.
- The cuboctahedron.
- Prisms and many of the prismatoids
- Since being transparent is invariant under affine transformations, also parallelepipeds etc.
The following polytopes are not transparent:
- Regular polygons with $\ge 5$ vertices.
- The icosahedron, dodecahedron and icosidodecahedron.