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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?

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

--- rk

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    $\begingroup$ I don't think that this answer is correct. For example the convex hull of the six points $(0,0,0,0),(-1,0,0,0),(0,-1,0,0),(0,0,1,0),(0,0,0,1),(1,1,1,1)$ in $\mathbb{R}^4$ has $9$ facets whereas the bipyramid on the $3$-simplex has only $8$ facets. $\endgroup$
    – Hans
    Feb 9, 2018 at 15:36

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