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My question: how do we know that this is the unit cube and why is it denoted as [0,1]^n ? Why 2n inequalities and 2^n extreme points?

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If we write out each row of $Ix \le e$, we get $x_i \le 1$. There are $n$ such constraints.

Each row of $x \ge 0$ gives us $x_i \ge 0$. There are $n$ such constraints.

Hence we have $0 \le x_i \le 1$ for each $i \in \{1, \ldots, n\}$.

In total, we have $2n$ constraints as I have explained.

$[0,1]^n$ is the cartesian product of $[0, 1]$ $n$ times, it means every entry is in between $0$ and $1$.

To be a vertex, for each variable, it get to choose to be $0$ or $1$. Hence there are $2^n$ options in total.

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