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Suppose I have $n$ linearly independent vectors $v_1, \ldots, v_n$ in $\mathbb R^n$.

Is there an expression for the volume of the following set:

$$ \{x \in \mathbb R^n : |x \cdot v_i| \leq 1 \text{ for all } i\} $$

?

My understanding is that this is the polar set of $\{ \pm v_1, \ldots, \pm v_n\}$.

Alan C
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    This set is a parallelepiped similar to the one spanned by the $v_i$. –  May 30 '18 at 00:33
  • Ah! If $V$ is $n\times n$ and has $v_1, \ldots, v_n$ as its rows, and $w_1, \ldots, w_n$ denote the columns of $V^{-1}$, then the parallelepiped is generated by ${ \pm w_1, \ldots, \pm w_n}$. – Alan C May 30 '18 at 12:24

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