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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\lbra …

Here is a very simple way to show the positivity. Define $$f(d,x) = (1+x)^d/(1-x)^{d+1}.$$ Then, by induction $$ \frac{\partial^t f(d,x)}{\partial\, d^t} = f(d,x) \, \ln\biggl(\frac{1+x}{1-x}\biggr)^ …

(The first part of this is the same as @Fedor's answer; I just carried out his algorithm.) Each row $i$ can be written as $a_i^{(1)}x_i^{(1)}+a_i^{(2)}x_i^{(2)}$ where $x_i^{(1)},x_i^{(2)}$ are non-ne …

The book Combinatorial Matrix Classes by Richard Brualdi has a number of generalizations of a combinatorial nature, around Chapter 8. Also, the least common denominator of the vertices of a rational …

Let $2\le k\le n-1$ and define the polytope $$P_k(n) = \lbrace (x_1,\ldots,x_n) \in \mathbb{R}^n : -1\le x_{i_1}+\cdots +x_{i_k} \le 1 \text{ for all } 1\le i_1\lt\cdots\lt i_k\le n\rbrace.$$ There …

Let $M(n,k)$ be the set of $n\times n$ matrices of nonnegative integers such that every row and every column sums to $k$. Let $P(n,k)$ be the fraction of such matrices which have no zero entries, equ …

I'll complement Christian's answer with an example in the other direction. Consider the polytope of $8\times 8$ symmetric doubly-stochastic matrices with 0 diagonal. The period of the Ehrhart quasipol …

A complete solution with references can be found in Section 8.1 of Brualdi, Combinatorial Matrix Classes, Cambridge University Press, 2006. Here is how to make an extreme point, and all extreme points …