Simplicial polytope in $\mathbb{R}^n$ with $n+2$ vertices

Question

I am interested in simplicial polytopes of dimension $n$ with exactly $n+2$ vertices. Is there a nice characterization of those? For $n=2$ there is of course only the quadrilateral but what about in arbitrary dimensions? Are the $f$-vectors known?

Answer 1

Those are the bipyramids on the n-1-dimensional simplices.
From that fact then you can deduce the f-vectors directly.

--- rk