delaunay tessellations of point lattices - freie universität · pdf filedelaunay...
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Delaunay Tessellations of Point Lattices Theory, Algorithms and Applications
Achill Schürmann(University of Rostock)
( based on work with Mathieu Dutour Sikiric and Frank Vallentin )
Berlin, October 2013
ERC Workshop Delaunay Geometry
Polytopes, Triangulations and Spheres
Делоне
Boris N. Delone1890-1980
( Delone )French: Delaunay
Delone tessellations
Delaunay subdivisions of lattices
L = hexagonal lattice 2-periodic (m=2)
Delone star and DV-cell
( ) =�
∈ R : � � ≤ � − � ∈�
Up to translation, there is only on DV-cell in a lattice:
Delone star and DV-cell
( ) =�
∈ R : � � ≤ � − � ∈�
Up to translation, there is only on DV-cell in a lattice:
all Delone polyhedra incident to a given vertex
It is dual to a Delone star
How do I compute
the DV-cell(or Delone star)?
Computing DV-cells(first approach)
THM (Voronoi, 1908):
∈ ( )± �= +
Computing DV-cells(first approach)
THM (Voronoi, 1908):
⇒ ( ) ( − )
∈ ( )± �= +
Computing DV-cells(first approach)
THM (Voronoi, 1908):
⇒ ( ) ( − )
0
∈ ( )± �= +
Computing DV-cells(first approach)
THM (Voronoi, 1908):
⇒ ( ) ( − )
0
PLAN
• compute facets
• obtain vertices
∈ ( )± �= +
Computing DV-cells
STEP 1: Compute an initial vertex (an initial Delone Polyhedron)
(practical approach)
Computing DV-cells
STEP 1: Compute an initial vertex (an initial Delone Polyhedron)
(practical approach)
Computing DV-cells
STEP 1: Compute an initial vertex (an initial Delone Polyhedron)
(practical approach)
Computing DV-cells (continued...)
STEP x: Compute neighboring vertices (neighboring Delone Polyhedra)
• Compute facets of Delone polyhedron • For each facet find neighboring Delone polyhedron• Test if they are equivalent to one of the known ones
Computing DV-cells (continued...)
STEP x: Compute neighboring vertices (neighboring Delone Polyhedra)
• Compute facets of Delone polyhedron • For each facet find neighboring Delone polyhedron• Test if they are equivalent to one of the known ones
Computing DV-cells (continued...)
STEP x: Compute neighboring vertices (neighboring Delone Polyhedra)
• Compute facets of Delone polyhedron • For each facet find neighboring Delone polyhedron• Test if they are equivalent to one of the known ones
( up to translation, central inversion, ...)
Computing DV-cells (continued...)
STEP x: Compute neighboring vertices (neighboring Delone Polyhedra)
• Compute facets of Delone polyhedron • For each facet find neighboring Delone polyhedron• Test if they are equivalent to one of the known ones
( up to translation, central inversion, ...)
Well suited for exploiting symmetry!
Computational Results
obtained by Mathieu using polyhedral
Mathieu
Computational Results
obtained by Mathieu using polyhedral
Mathieu
IN: Complexity and algorithms for computing Voronoi cells of lattices, Math. Comp. (2009)
• computation of vertices for many different DV-cells of lattices (in particular for Coxeter-, Laminated and Cut-Lattices) • verified that Leech Lattice cell has 307 vertex orbits (Conway, Borcherds, et. al.)
Application I: Covering Constants
Application I: Covering Constants
µ( ) = sup∈ ( )
( )
Application II: Quantizer Constants
What happensif we vary the lattice?
Two views
Instead of varying the lattice we can equivalently vary the norm
Z � � =√
=
= Z � � =√
⇔
Quadratic Forms
Arithmetical Equivalence
Dictionary
Dictionary
Delone tessellations revisited
Delone tessellations revisited
Secondary cones
D R Z
∆(D) =�
∈ S> : ( ) = D�
Secondary cones
D R Z
∆(D) =�
∈ S> : ( ) = D�
Baranovskii Cones
Z
∆( ) =�
∈ S> : ∈ ( )�
DEF:
Baranovskii Cones
Z
∆( ) =�
∈ S> : ∈ ( )�
DEF:
THM:∆( )
Baranovskii Cones
Z
∆( ) =�
∈ S> : ∈ ( )�
DEF:
THM:∆( )
D ∆(D) =�
∈D∆(P)
Note:
Baranovskii Cones
Z
∆( ) =�
∈ S> : ∈ ( )�
DEF:
THM:∆( )
D ∆(D) =�
∈D∆(P)
Note:
⇒ ∆(D)
Application:Finding best lattice coverings
Application:Finding best lattice coverings
�∈ S> : ( ( )) ≤
�
THM (Barnes, Dickson; 1968): Among PQFs in the closure of a secondary cone
Θ
Application:Finding best lattice coverings
�∈ S> : ( ( )) ≤
�
THM (Barnes, Dickson; 1968): Among PQFs in the closure of a secondary cone
Θ
Note:
Voronoi’s second reduction
Voronoi’s second reduction
THM (Voronoi, 1908):
• there exist finitely many inequivalent secondary cones
• inclusion of faces corresponds to coarsening of subdivisions
• closures of secondary cones tesselate S>
Voronoi’s second reduction
THM (Voronoi, 1908):
• there exist finitely many inequivalent secondary cones
• inclusion of faces corresponds to coarsening of subdivisions
• closures of secondary cones tesselate S>
=> top-dimensional cones come from triangulations
Voronoi’s second reduction
THM (Voronoi, 1908):
• there exist finitely many inequivalent secondary cones
• inclusion of faces corresponds to coarsening of subdivisions
• closures of secondary cones tesselate S>
=> top-dimensional cones come from triangulations
Already known...
Already known...
IDEA: In higher dimensions, determine the best lattice coverings with a given group of symmetries!? (obtaining all Delone subdivisons with a given symmetry)
G-Theory?
G-Theory?
G-Theory?
IDEA: Intersect secondary cones with a linear subspace T
G-Theory?
IDEA: Intersect secondary cones with a linear subspace T
DEF: ∩∆(D)
T-secondary cones• T-secondary cones tesselate S> ∩
T-secondary cones• T-secondary cones tesselate S> ∩
�
:⇔ ∃ ∈ (Z) = � ⊆DEF:
T-secondary cones• T-secondary cones tesselate S> ∩
�
:⇔ ∃ ∈ (Z) = � ⊆DEF:
THM: ⊂ (Z)
T-secondary cones• T-secondary cones tesselate S> ∩
�
:⇔ ∃ ∈ (Z) = � ⊆DEF:
• Delone subdivision of a neighboring T-secondary cone can be obtained by a T-flip in repartitioning polytopes
THM: ⊂ (Z)
T-secondary cones• T-secondary cones tesselate S> ∩
�
:⇔ ∃ ∈ (Z) = � ⊆DEF:
• Delone subdivision of a neighboring T-secondary cone can be obtained by a T-flip in repartitioning polytopes
THM: ⊂ (Z)
Application to Lattice Coverings
What about the nice lattices?
What about the nice lattices?
Prominent Example:
(E )
Θ
What about the nice lattices?
Question: Are there local maxima??
Prominent Example:
(E )
Θ
What about the nice lattices?
Question: Are there local maxima??
YES! E Θ
Prominent Example:
(E )
Θ
What about the nice lattices?
Question: Are there local maxima??
YES! E Θ= , . . . ,
Prominent Example:
(E )
Θ
What about the nice lattices?
Question: Are there local maxima??
THM: A necessary condition for a local maximum is thatevery Delone polytope attaining the covering radius
is an extreme Delone polytopedim∆( ) =
YES! E Θ= , . . . ,
Prominent Example:
(E )
Θ
i-eutaxy and i-perfectness
DEF: Q is i-perfect if
DEF: Q is i-eutactic if
i-eutaxy and i-perfectness
DEF: Q is i-perfect if
DEF: Q is i-eutactic if
THM: Θ⇔
Bahavior of nice lattices
lattice covering densityZ global minimumA2 global minimumD4 almost local maximumE6 local maximumE7 local maximumE8 almost local maximumK12 almost local maximumBW16 local maximumΛ24 local minimum
Application: Minkowski Conjecture
Conjecture:
( ) = | · · · | ⊂ R det =
sup∈R
inf∈
( − ) ≤ −
= ( , . . . , )Z ( ) =
≤
Covering Conjecture
≤
Covering Conjecture
Local covering maxima among well rounded lattices are attained by T-extreme Delone Polyhedra and there are only
finitely many of them in every dimension.(with T = space of well rounded lattices)
THM:
≤
References
• Complexity and algorithms for computing Voronoi cells of lattices, Math. Comp. (2009)
• computation of covering radius and Delone subdivisions for many lattices• verified that Leech lattice cell has 307 vertex orbits (Conway, Borcherds, et. al.)
• A generalization of Voronoi’s reduction theory and its application, Duke Math. J. (2008)
• Inhomogeneous extreme forms, Annales de l'institut Fourier (2012)
• generalized Voronoi’s reduction for L-type domains to a G- and T-invariant setting• obtained new best known covering lattices and classified totally real thin number fields
• characterization of locally extreme forms for the sphere covering problem
http://www.geometrie.uni-rostock.de