TITLE: Name of a polytope
QUESTION [16 upvotes]: What is the name of the polytope $\Sigma\cap (-\Sigma)$ for $\Sigma$ a $d-$simplex with barycenter at the origin?
In dimension $2$, one gets a hexagon, in dimension $3$ an octahedron (given by the $6$ midpoints of edges), in dimension $4$
a polytope with 30 vertices (given for example by all permutations of $(0,1,1,-1,-1)$).
(More generally, this polytop has ${2n\choose n}$ vertices in dimension $d=2n-1$
and $(2n+1){2n\choose n}$ vertices in dimension $d=2n$.)

REPLY [5 votes]: I do not know of any specific name for this family of polytopes, but their Coxeter diagrams are of a recognizable form.
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In fact, every second one is a hyper-simplex. In general, you can look them up on Wikipedia's list of uniform polytopes.
The intersection $\Sigma \cap(-\Sigma)$ of $d$-dimensional simplices can be constructed as
$$\Big\{\sum_{i=1}^{d+1} \lambda_i e_i \in\Bbb R^{d+1} \;\Big|\;  \sum_{i=1}^{d+1}\lambda_i=1,\lambda_i\in\big[0,\frac{2}{d+1}\big]\Big\}.$$
This is almost the convex hull of the $e_i$, but the coefficients of the convex combinations are more restricted.
With this, it is possible to prove what Vince's suspected about the coordinates of the vertices.