Convex and discrete geometry Lattices and geometry of numbers Conjecture

Voronoi's conjecture on parallelohedra

Builds on Parallelohedron Voronoi cell Lattice (group)

Statement

Let $d \ge 1$ and let $P \subset \R^d$ be a parallelohedron. Then there are a lattice $\Lambda \subset \R^d$ and an affine bijection $A \colon \R^d \to \R^d$ such that $A(P)$ is the Voronoi cell of $\Lambda$ .

Remarks

The converse holds: the Voronoi cell of every lattice is a parallelohedron. Voronoi proved the conjecture for primitive parallelohedra, those whose tiling has exactly $d + 1$ tiles at each vertex. Delone proved it in dimensions $d \le 4$ . Erdahl proved it for parallelohedra that are zonotopes. The general case is open.

Reading path

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Lattice (group) concept
Parallelohedron concept
Voronoi cell concept
Voronoi's conjecture on parallelohedra conjecture

Tag	`F03A4MKA5V`
URI	`m2://F03A4MKA5V`
Kind	conjecture
Writing	W0 stub
MSC	52C22, 52B11, 11H06
Status	This copy of the site has no build data. Run `m2 status F03A4MKA5V`.
About	Parallelohedron
Uses	Parallelohedron, Voronoi cell, Lattice (group)

Voronoi's conjecture on parallelohedra

Statement

Remarks

Reading path

Further reading

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