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Definition 1. Given a body $V$ in $\mathbb R^n$, the function $p_V\colon \mathbb R_+\to \mathbb R_+$ $$p_V(r)=\mathop{\rm vol} [V\cap B_r(0)]$$ will be called profile of $V$.

Definition 2. Define Voronoi cell of lattice $L$ in $\mathbb R^n$ as $$V_L=\{\,x\in \mathbb R^n;\,|x|\le |x+\ell| \ \text{for any}\ \ell\in L\,\}.$$

Question. Can it happen that Voronoi cells of a pair of lattices have the same profile, but not isometric?

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    $\begingroup$ The $II_{16,0}$ and $E_8 + E_8$ lattices in dimension 16 have different Voronoi domains, since the reflection planes for the roots yield different Dynkin diagrams. I think the domains have the same profile, because the theta constants of the lattices are equal. $\endgroup$
    – S. Carnahan ♦
    May 20, 2011 at 2:21
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    $\begingroup$ Equality of theta-functions is equivalent to the fact that both latices have the same number of points in any ball centered at the origin (?). I do not see why this property related to the one I want --- they sound similar, but I do not see a bridge between them. $\endgroup$
    – Anton Petrunin
    May 20, 2011 at 3:15
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    $\begingroup$ @Will Jagy: There are quite a few polytopes with the same profile, even centrally symmetric ones. You can stellate two pairs of opposite sides of an icosahedron (or octagon), getting the same profile regardless of the pairs of opposite sides you choose. The symmetries do not act transitively on the possibilities. However, it's much easier to motivate considering the profile of the Voronoi domain of a lattice since it tells you the distribution of distances to the lattice in $\mathbb{R}^n / L$. $\endgroup$
    – Douglas Zare
    May 20, 2011 at 17:24
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    $\begingroup$ @Scott Carnahan: Given a lattice, one can consider the norm of the "most distant point to the lattice" that is closest to the origin. Is that invariant the same for E_8⊕E_8 and for II_{16,0}? Let me rephrase because I might have said things in a confusing way: I'm asking for the diameters (or maybe "radius") of the Voronoi cells: are they the same for those two lattices? $\endgroup$
    – André Henriques
    Jul 20, 2011 at 23:41
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    $\begingroup$ Also known as the "covering radius" of the lattice. $\endgroup$
    – Noam D. Elkies
    Jan 27, 2014 at 1:14

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My apologies for a trivial comment (not an answer), well-understood by the OP. But perhaps this illustration will inspire advancement on this long-unsolved question, and invite clarification if I am misinterpreting.

I abandon the requirement that $V$ be the Voronoi cell of a lattice, and just look at convex bodies:
   TruncSquareProfile
Here the two shapes have the same "profile" in Anton's sense: they intersect origin-centered balls in the same volumes. But they are not congruent, not equivalent under an isometry. So the challenge is to arrange something similar for lattice Voronoi cells.

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  • $\begingroup$ Since a Voronoi cell of a lattice set is an intersection of closed half spaces, then these convex sets are Voronoi cells of some lattice sets. $\endgroup$
    – Hee Kwon Lee
    Feb 3, 2019 at 16:47

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