mathieu dutour sikiric, achill schuermann and frank vallentin- parameter space of delaunay...

8/3/2019 Mathieu Dutour Sikiric, Achill Schuermann and Frank Vallentin- Parameter Space of Delaunay tessellations: L-types

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Parameter Space of Delaunay tessellations:

L-types

Mathieu Dutour SikiricInstitute Rudjer Boskovic

Achill SchuermannMagdeburg University

Frank Vallentin

CWI Amsterdam

November 24, 2011


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Gram matrix and lattices

What really matters for lattice is their isometry class, i.e., if uis an isometry ofRn then the lattices L and uL have the samegeometry.

Denote Sn>0 the cone of real symmetric positive definite n n

matrices and Sn

0 the positive semidefinite ones. Lattice L spanned by v1, . . . , vn corresponds to

Gv = (vi, vj)1i,jn Sn>0 .

Gv depends only on the isometry class of L. Given M Sn

>0, one can find vectors v1, . . . , vn such thatM = Gv.


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Empty sphere and Delaunay polytopes

A sphere S(c, r) of radius r and center c in an n-dimensional

lattice L is said to be an empty sphere if:(i) v c r for all v L,

(ii) the set S(c, r) L contains n + 1 affinely independent points.

A Delaunay polytope P in a lattice L is a polytope, whose

vertex-set is L S(c, r).

c

r


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Equalities and inequalities

Take M = Gv with v = (v1, . . . , vn) a basis of lattice L.

If V = (w1, . . . , wN) with wi Zn are the vertices of a

Delaunay polytope of empty sphere S(c, r) then:

||wi c|| = r i.e. wTi Mwi 2w

Ti Mc + c

TMc = r2

Substracting one obtains

{wTi Mwi wTj Mwj} 2{wTi wTj }Mc = 0

Inverting matrices, one obtains Mc = (M) with linear andso one gets linear equalities on M.

Similarly ||w c|| r translates into linear inequalities on M:Take V = (v0, . . . , vn) a simplex (vi Zn), w Zn. If onewrites w =

ni=0 ivi with 1 =

ni=0 i, then one has

||w c|| r wTMw n

i=0

ivTi Mvi 0


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III. L-type

domain


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L-type domains

Take a lattice L and select a basis v1, . . . , vn.

We want to assign the Delaunay polytopes of a lattice.

Geometrically, this means that

1v

2

v

2v1

v

are part of the same L-type domain.

A L-type domain is the assignment of Delaunay polytopes, soit is also the assignment of the Voronoi polytope of the lattice.

Primitive L-types

If one takes a generic matrix M in Sn>0, then all its Delaunay

are simplices and so no linear equality are implied on M.

Hence the corresponding L-type is of dimension n(n+1)

2

, theyare called primitive


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Equivalence and enumeration

Voronois theorem The inequalities obtained by taking

adjacent simplices suffice to describe all inequalities. The group GLn(Z) acts on S

n>0 by arithmetic equivalence and

preserve the primitive L-type domains.

Voronoi proved that after this action, there is a finite number

of primitive L-type domains. Bistellar flipping creates new triangulation. In dim. 2:

Enumerating primitive L-types is done classicaly: Find one primitive L-type domain. Find the adjacent ones and reduce by arithmetic equivalence.


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The partition ofS2>0 R3

If q(x, y) = ux2 + 2vxy + wy2 then q S2>0 if and only if

v2 < uw and u > 0.

w

v

u


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We cut by the plane u + w = 1 and get a circle representation.

u

v

w


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Primitive L-types in S2>0:


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Rigid lattices

A lattice is rigid (notion introduced by Baranovski & Grishukhin) if

its L-type domain has dimension 1.

One rigid in dimension 1: Z.

No rigid lattices in dimension 2 and 3.

one rigid lattice in dimension 4: it is D4. 7 rigid lattices in dimension 5.

E. Baranovskii, V. Grishukhin, Non-rigidity degree of a latticeand rigid lattices, European J. Combin. 22-7 (2001) 921935.

In dimension 6, we obtained 25263 rigid lattices. Probably

many more. M. Dutour and F. Vallentin, Some six-dimensional rigid

lattices, Proceedings of Third Vorono Conference of theNumber Theory and Spatial Tesselations, 102108.


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Enumeration ofL-types

Dimension Nr. L-type Nr. primitive Nr rigid lattices

1 1 1 12 2 1 03 5 1 0

Fedorov Fedorov

4 52 3 1DeSh Voronoi

5 179377 222 7Engel BaRy, Engel & Gr BaGr

6 ? 2.5.106 2.104

Engel, Va DuVa

7 ? ? ?


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IV. Covering

andoptimization


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Lattice covering

We consider covering ofRn by n-dimensional balls of thesame radius, whose center belong to a lattice L.

The covering density has the expression

(L) =(L)nndet(L)

1

with (L) being the largest radius of Delaunay polytopes andn the volume of the unit ball B

n.

Objective is to minimize (L). Solution for n 5: An.


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Lattice packing-covering

L is a n-dimensional lattice.

We want a lattice, such that the sphere packing (resp,covering) obtained by taking spheres centered in L withmaximal (resp, minimal) radius are both good.

The quantity of interest is

(L)(L) = { (L)(L) }n 1

Lattice packing-covering problem: minimize (L)(L) .

Dimension Solution

2 A2 4 H4 (Horvath lattice)3 A3 5 H5 (Horvath lattice)

J. Horvath, PhD thesis: Several problems of n-dimensionalgeometry, Steklov Inst. Math., 1986.


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Optimization problem

We want to find the best covering, packing-covering.

Thm. Given a L-type domain LT, there exist a unique lattice,which minimize the covering density over LT.

Thm. Given a L-type domain LT, there exist a lattice

(possibly several), which minimize the packing-coveringdensity over LT.

See for more details

A. Schurmann and F. Vallentin, Computational approaches tolattice packing and covering problems, Discrete Comput.Geom.35-1 (2006) 73116.


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Radius of Delaunay polytope

Fix a primitive L-type domain, i.e. a collection of simplexes asDelaunay polytopes D

1, . . . , Dm.

Thm. For every Di = Conv(0, v1, . . . , vn), the radius of theDelaunay polytope is at most 1 if and only if

4 v1, v1 v2, v2 . . . vn, vn

v1, v1 v1, v1 v1, v2 . . . v1, vnv2, v2 v2, v1 v2, v2 . . . v2, vn...

......

. . ....

vn, vn vn, v1 vn, v2 . . . vn, vn

Sn+10

It is a semidefinite condition. B.N. Delone, N.P. Dolbilin, S.S. Ryskov and M.I. Stogrin, A

new construction of the theory of lattice coverings of an

n-dimensional space by congruent balls, Izv. Akad. NaukSSSR Ser. Mat. 34 (1970) 289298.

C


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Convex programming problems

A convex programming problem is of the following typeMaximize g over C with g convex and C convex.

For many subproblems, there are efficient procedures: quadratically constrained quadratic programming, geometrical programming, approximation in lp-norms, optimization over the cone of positive semidefinite symmetric

matrices, finding extremal ellipsoids, etc.

See for more details:

Y. Nesterov, A. Nemirovskii, Interior-point polynomialalgorithms in convex programming, (1994) SIAM Studies inApplied Mathematics, 13.

C i bl


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Covering problem

Fix a primitive L-type domain, i.e. a collection of simplexes asDelaunay polytopes D1, . . . , Dm.

Minkowski The function log det(M) is strictly convex onSn>0.

Solve the problem M in the L-type (linear condition), the Delaunay Di have radius at most 1 (semidefinite

condition), minimize log det(M) (strictly convex).

The above problem is solved by the interior point methods

implemented in MAXDET by Vandenberghe, Boyd & Wu.

Unicity comes from the strict convexity of the objectivefunction.

P ki i bl


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Packing covering problem

We fix a primitive L-type domain.

A shortest vector is an edge of a Delaunay. So, from theDelaunay decomposition, we know which vectors v1, . . . , vpcan be shortest.

We consider the problem on (M, m) Sn>0 R M belong to the L-type domain (linear constraint) all Delaunay have radius at most 1 (semidefinite condition) m ||vj||

2 = vtj Mvj for all i (linear constraint) maximize m.

The maximal value of m gives the maximal length of shortest

vector and so the best packing-covering over a specificprimitive L-type domain.

A priori no unicity.


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V. L-types

ofSn>0-spaces

Sn


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Sn>0-spaces

A Sn>0-space is a vector space SP of S

n

, which intersect Sn>0.

We want to describe the Delaunay decomposition of matricesM Sn>0 SP.

Motivations: The enumeration of L-types is done up to dimension 5, perhaps

possible for dimension 6 but certainly not for higher dimension. We hope to find some good covering, and packing-covering by

selecting judicious SP. This is a search for best but unprovento be optimal coverings.

A L-type in SP is an open convex polyhedral set included inSn>0 SP, for which every element has the same Delaunay

decomposition.

Rigidit a d i iti it


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Rigidity and primitivity

(SP, L)-types form a polyhedral tessellation of the spaceSP Sn>0.

If M SP Sn>0, then the rigidity degree of M is the

dimension of the smallest L-type containing M, it is computedusing the Delaunay decomposition of M.

A (SP, L)-type is primitive if it is full-dimensional in SP. A (SP, L)-type is rigid if it is one dimensional. Las Vegas algorithm: Algorithm for finding a primitive

(SP, L)-type domain

Generate a random element in Sn>0 SP. Compute its Delaunay decomposition.

If the dimension of the (SP, L)-type is maximal, then return it. Otherwise, rerun.

Testing equivalence of (SP L) type


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Testing equivalence of (SP, L)-type

Given a primitive (SP, L)-type domain LT, find its extremerays ei and normalize the corresponding matrices by imposingthat they have integer coefficients with gcd = 1.

We associate to the (SP, L)-type LT the matrix inSn

>0

SP: MLT = i

ei

Two primitive (SP, L)-type domains LT1 and LT2 areisomorphic if there a matrix P such that

PMLT1tP = MLT2 and PSP

tP = SP

First equation is solved by program Isom and we iterate overthe possible solutions for testing the second.

Lifted Delaunay decomposition


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Lifted Delaunay decomposition

The Delaunay polytopes of a lattice L correspond to the

facets of the convex cone C(L) with vertex-set:

{(x, ||x||2) with x L} Rd+1 .

Flipping


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Flipping

Take a primitive (SP, L)-type domain with Delaunaypolytopes D1, . . . , Dp.

If F is a facet of Di and D is the other Delaunay polytope,

then it defines an inequality fD,Di(M) 0. This form a finiteset of defining inequalities of the (SP, L)-type.

We can extract relevant inequalities, which correspond tofacets of the (SP, L)-type. Select a relevant ineq. f(M) 0.

One has f(M) = 1fDj(1),D1 (M) = = rfDj(r),Dr(M) for

some i > 0 and some Delaunay Di adjacent to Dj(i) on a

facet Fi. If one moves to f(M) = 0, then all Fi disappear and the

corresponding Delaunays merge.

Geometrical expression


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Geometrical expression

The glued Delaunay form a Delaunay decomposition for a

matrix M in the (SP, L)-type satisfying to f(M) = 0. The flipping break those Delaunays in a different way.

Two triangulations ofZ2 correpond in the lifting to:

The polytope represented is called the repartitioning polytope.

Flipping of a repartitioning polytope


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Flipping of a repartitioning polytope

Given a Delaunay decomposition D, the graph G(D) isformed of all Delaunays with two Delaunay di, dj adjacent if:

di and dj share a facet the inequality fdi,dj(M) = f(M) for > 0

For every connected component C of this graph, the

repartitioning polytope R(P) is the polytope with vertex set

{(v, tvMv) with v a vertex of a Delaunay of C}

Combinatorially flipping correspond to switching from the

lower facets to the higher facets of the lifted merging ofDelaunay polytopes.

Enumeration technique


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Enumeration technique

Find a primitive (SP, L)-type domain, insert it to the list asundone.

Iterate

For every undone primitive (SP, L)-type domain, compute thefacets. Eliminate redundant inequalities. For every non-redundant inequality realize the flipping, i.e.

compute the adjacent primitive (SP, L)-type domain. If it isnew, then add to the list as undone.


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VI. Applications

Space of invariant forms


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Space of invariant forms

Given a subgroup G of GLn(Z), define

SP(G) =

X Sn such that gXtg = X for all g G

Given a Sn>0-space SP, define

Aut(SP) =

g GLn(Z) such that

gXtg = X for all X SP

A Bravais group satisfies to Aut(SP(G)) = G.

Equivariant L-type domains


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Equivariant L type domains

Equivariant L-type domains are L-types of a Sn>0-space

SP(G) for G Bravais. Thm. (Zassenhaus) One has the equality

{g GLn(Z) | gSP(G)tg = SP(G)} = NGLn(Z)(G)

Thm. For a given finite group G GLn(Z), there are a finitenumber of equivariant L-types under the action of NGLn(Z)(G).

Note that if a T-space SP is defined by rational equations,then it does not necessarily have a finite number of L-types

underAut

(SP).Example (courtesy of Yves Benoist):

SP = R(x2 + 2y2 + z2) + R(xy)

Small dimensions


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Dimension 6: Vallentin found a better lattice covering than A6 in the vicinity

of E

6 . No better in Bravais groups of rank 4.

Dimension 7: Vallentin & Schurmann found a better lattice covering than A7

in the vicinity of E7 .

No better in Bravais groups of rank 4. Dimension 8:

Vallentin & Schurmann proved that E8 is not a local optimumof the covering density.

A. Schurmann and F. Vallentin, Local covering optimality of

lattices: Leech lattice versus root lattice E8, Int. Math. Res.Not. 2005, no. 32, 19371955.

Conjecture (Zong) E8 is the best lattice packing-covering indimension 8.

C. Zong, From deep holes to free planes, Bull. Amer. Math.Soc. (N.S.) 39-4 (2002) 533555.

Extension of Coxeter lattices


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Anzin & Baranovski computed the Delaunay decompositionsof the lattices A59, A

411, A

713, A

514, A

815 and found them to be

better coverings than An.

We do extension along short vectors

or compute in the Bravais space of short vectors. We manage to find record coverings in dimension 9, 11, 13,

14 and 15.

Lamination


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Given a n-dim. lattice L, create a n + 1-dim. lattice L:

c

h

L

vn+1

n+1v +L

The point c is fixed and we adjust the value of h.

This defines a T-space.

Lamination


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In terms of Gram matrices

Gram(L) = A and Gram(L) = A Atc

cA

c is the projection of the vector (0, . . . , 0, 1) on the lattice L. The symmetries of L are the symmetries of L preserving the

center c and if 2c Zn the othogonal symmetry

In 02c 1

c can be chosen as center of a Delaunay.

For the covering problem things are not so simple. One cannot solve the general problem with c unspecified, since

it has no symmetry and too much parameters One restriction is to assume the value of c, this makes a rank

2 problem.

Doing lamination over A59 and A411 one gets a record covering

in dimension 10 and 12.

Best known lattice coverings


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d lattice covering density

1 Z1 1 13 Lc13 7.7621082 A2 1.209199 14 L

c14 8.825210

3 A3 1.463505 15 Lc15 11.004951

4 A4 1.765529 16 A

16 15.3109275 A5 2.124286 17 A

917 12.357468

6 Lc6 2.464801 18 A

18 21.8409497 Lc7 2.900024 19 A

1019 21.229200

8 Lc8 3.142202 20 A720 20.366828

9 Lc9 4.268575 21 A1121 27.773140

10 Lc10 5.154463 22

22 27.883911 Lc11 5.505591 23

23 15.3218

12 Lc

12 7.465518 24 Leech 7.903536


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mathieu dutour sikiric, achill schuermann and frank vallentin- parameter space of delaunay...

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