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Suppose $\Lambda$ is a $3$ dimensional lattice inside $\mathbb{R}^4$ and let $E$ be the subspace $\mathbb{R}$-spanned by $\Lambda$.

What exactly is meant by the unit ball in $E$? This is something that came up in a research paper I was reading and I wasn't sure what it meant... any clarification is appreciated. thank you

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    $\begingroup$ Is there any reason that it shouldn't be the obvious one? (The intersection of of a unit ball (of $\mathbb{R}^4$), centered at $p\in E$, with the subspace $E$.) $\endgroup$
    – Willie Wong
    Commented Aug 24, 2021 at 16:40
  • $\begingroup$ to me this was not too obvious... is it easy to deduce that the volume would be independent of the choice of $p \in E$? or is $p$ a special point? $\endgroup$
    – Johnny T.
    Commented Aug 24, 2021 at 16:46
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    $\begingroup$ If your notion of volume is derived from a translation invariant measure, then yes. In your case, I guess the 3 dimensional Hausdorff measure is a natural choice. // I would suggest you include more context in your question. Perhaps pointing to the paper that you are reading that is bothering you. $\endgroup$
    – Willie Wong
    Commented Aug 24, 2021 at 18:25

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