Question:Suppose $\{v_1,\ldots,v_{n-1}\}$ is an integer basis for the lattice $$L=\{(x_1,\ldots,x_n)\in\mathbb{Z}^n: > a_1x_1+\cdots+a_nx_n=0\},$$ where the $a_i$ are fixed nonzero integers. Is the volume $V(P)=\det(L)$ (see this for a proof that they are equal) of its fundamental parallelotope $P=\{t_1v_1+\cdots+t_{n-1}v_{n-1} > \mid t_i\in[0,1)\}$ necessarily equal to $$\frac{\sqrt{a_1^2+\cdots+a_n^2}}{\gcd(a_1,\ldots,a_n)}?$$

I used the case $n=3$ along with Minkowski's theorem (in the geometry of numbers) to solve the following Miklos problem from 2000:

Let $a<b<c$ be positive integers. Prove that there exist integers $x,y,z$, not all zero, such that $ax+by+cz=0$ and $\max(|x|,|y|,|z|)\le > 1+\frac{2}{\sqrt3}\sqrt{c}$, and show that the constant $\frac{2}{\sqrt3}$ cannot be improved.

However, I was only able to find a brute force proof for this special case (see lemma 1 in my AoPS post here), and I'm not sure if it's as easy for larger values of $n$.

But I'm pretty sure this should be true in general (I've tried several cases for $n=4$ and $n=5$), so I would appreciate it if someone could give a (clean?) proof, reference, or counterexample. Thanks!

QED. $\endgroup$ – Noam D. Elkies Jul 26 '12 at 3:53primitivelyembedded in ${\bf Z}^n$; that is, if $L_0$ contains $mv$ for some nonzero $m\in\bf Z$ and $v \in {\bf Z}^n$ then $L_0$ contains $v$. Let $L \subset {\bf Z}^n$ consist of the integer vectors orthogonal to every vector in $L_0$. Then the fundamental parallelotopes of the lattices $L_0$, $L$ have the same volume. Victor's question is the special case when $L_0$ is the rank-$1$ lattice ${\bf Z}\cdot(a_1,\ldots,a_n)$. $\endgroup$ – Noam D. Elkies Jul 26 '12 at 4:53