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Let $b\gt a\gt 0$ be constants. Define $P_n(a,b)$ to be the set of all $(x_1,\ldots,x_n)\in\mathbb{R}^n$ satisfying $$ |c_1 x_1 + \cdots + c_n x_n| \le 1$$ for every choice of $c_1,\ldots,c_n\in\lbrace a,b\rbrace$.

This is a finite convex polytope and I wonder if it has been studied and perhaps has a name.

The vertices are rather few, $n(n+1)$ in total, consisting of all permutations of the points $$ \Bigl(-\frac{1}{b}, 0,\ldots,0\Bigr),~~ \Bigl(\frac{1}{b}, 0,\ldots,0\Bigr),~~ \Bigl(-\frac{1}{b-a},\frac{1}{b-a},0,\ldots,0\Bigr). $$

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  • $\begingroup$ Dear Brendan, it looks like a relative of the "hypersimplex" (in this case) the convex hull of centers of edges of a simplex. (More generally, centers of $k$-faces.) $\endgroup$
    – Gil Kalai
    Commented Jan 29, 2016 at 19:05

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I've seen some papers where the convex hull of the roots in a root lattice is called the root polytope of that root system. In type $A_n$ this is the convex hull in $\mathbb R^{n+1}$ of the vectors $e_i-e_j, 1\le i,j\le n+1$. These lie in the hyperplane $\sum_{i=1}^{n+1} x_i=0$. If you project to $\mathbb R^n$ by $x_i'=x_i+\frac{a}{b}x_{n+1}$ you get your polytope.

For example, the f-vector and some other stuff about the root polytope of type $A_n$ was done in "Root polytopes and growth series of root lattices" by Ardila, Beck, Hosten, Pfeifle, and Seashore.

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