Let P(d,v) denote the number of convex polytopes (up to combinatorial equivalence) in $\,$d dimensions with v vertices. The octahedron and triangular prism are two familiar examples with d=3 and v=6 . There are five others bringing the total to P(3,6) = 7.
In four dimensions, any polytope with 6 vertices has just one more than the bare minimum needed to form a simplex. $\,$ Apparently there are P(4,6) = 4$\,$ such.
For completeness, there is only the hexagon if d=2 and only the simplex when d=5.$\,$ Therefore, the full count of all polytopes with v=6 and $2\le d$$\,$$\le 5$ $\,$is given by $\,$(1,7,4,1).
Questions: (1) Is there a reference which shows illustrations for the seven polytopes in the$\,$ d=3, v=6 case?
(2)$\,$The set of 3 x 3 doubly stochastic matrices forms a four dimensional polytope inside$\,$ $\Bbb R^9$$\,$having six vertices (corresponding to the six permutation matrices). Is there a nice way to embed this in$\,$$\,$$\Bbb R^4$$\,$ or to understand it in some more direct geometric way?
(3)$\,$Suppose we fix$\,$ v $\,$the number of vertices and consider the sequence P(d,v) for $\,$$2 \le d$$\,$$ \le v-1$. $\,$ Is this sequence known or conjectured to be unimodal?
Thanks very much