Mathieu Dutour Sikiric, Konstantin Rybnikov, Achill Schuermann and Frank Vallentin- Covering maxima and Delaunay polytopes in lattices

8/3/2019 Mathieu Dutour Sikiric, Konstantin Rybnikov, Achill Schuermann and Frank Vallentin- Covering maxima and Delaunay polytopes in lattices

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Covering maxima and Delaunay polytopes in

lattices

Mathieu Dutour Sikiric

Rudjer Boskovic Institute,Croatia

Konstantin Rybnikov

Massachussets, Lowell, USA

Achill Schuermann

TU Delft, Netherland

Frank Vallentin

TU Delft, Netherland

January 16, 2010


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I. Quadratic functions

and the Erdahl cone


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The Erdahl cone

Denote by E2(n) the vector space of degree 2 polynomial

functions onRn

. We write f E2(n) in the form

f(x) = af + bf x + Qf[x]

with af R, bf Rn and Qf a n n symmetric matrix

The Erdahl cone is defined as

Erdahl(n) = {f E2(n) such that f(x) 0 for x Zn}

It is a convex cone, which is non-polyhedral since defined by

an infinity of inequalities. The group acting on Erdahl(n) is AGLn(Z), i.e. the group of

affine integral transformations

x b+ xP for b Zn and P GLn(Z)


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Scalar product

Definition: Iff, g

E

2(n

), then:

f, g = afag + bf, bg + Qf,Qg

Definition: For v Zn, define evv(x) = (1 + v x)2.

We have f, evv = f(v)

Thus finding the rays of Erdahl(n) is a dual descriptionproblem with an infinity of inequalities and infinite groupacting on it.

Definition: We also define

Erdahl>0(n) = {f Erdahl(n) : Qf positive definite}


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Empty sphere and Delaunay polytopes

Definition: A sphere S(c, r) of center c and radius r 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.

Definition: A Delaunay polytope P in a lattice L is apolytope, whose vertex-set is L S(c, r).

c

r

Delaunay polytopes define a tesselation of the Euclideanspace.


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Relation with Delaunay polytope

If D is a Delaunay polytope of a lattice L of empty sphereS(c, r) then for every basis v1, . . . , vn of L we define the

function

fD : Zn R

x = (x1, . . . , xn) n

i=1 xivi c2 r2

Clearly fD Erdahl>0(n). On the other hand if f Erdahl(n) then there exist a lattice

L of dimension k n, a Delaunay polytope D of L, afunction AGLn(Z) such that

f (x1, . . . , xn) = fD(x1, . . . , xk)

Thus the faces of Erdahl(n) correspond to the Delaunaypolytope of dimension at most n.

The rank of a Delaunay polytope is the dimension of the face

it defines in Erdahl(n).


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Perfect quadratic function

Definition: If f Erdahl(n) then

Z(f) = {v Zn

: f(v) = 0}

andDom f =

vZ(f)

R+evv

We have f,Dom f = 0. Definition: f Erdahl(n) is perfect if Dom f is of dimension

n+22

1.

This is equivalent to say that f defines an extreme ray in

Erdahl(n). The interval [0, 1] is a perfect Delaunay polytope and for every

n 6 there exist an n-dimensional perfect Delaunay polytope.

Geometrically, this means that the only affine transformationsthat preserve the Delaunay property are isometries and

homotheties.


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Eutacticity

If f Erdahl>0(n) then define f and cf such that

f(x) = Qf[x cf] f

Then define

uf(x) = (1 + cf x)2 +

f

nQ1f [x]

Definition: f Erdahl>0(n) is eutactic if uf relint(Dom f)and weakly eutactic if uf Dom f.

Definition: Take a Delaunay polytope P for a quadratic formQ of center cP and square radius P. P is called eutactic ifthere are v > 0 so that

1 =

vvertPv,

0 =

vvertPv(v cP),

Pn

Q1 = vvertP

v(v cP)(v cP)t.


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Strongly perfect Delaunay polytopes

Definition: We say that a finite, nonempty subset X in Rn

carries a spherical t-design if there is a similarity

transformation mapping X to points on the unit sphere Sn1so that

1

|Y|

yY

f(y) =1

n

Sn1

f(y)d(y).

holds, for all polynomials f R[x1, . . . , xn] up to degree t. If the vertices of P form a 2-design, then they define an

eutactic form.

Definition: A Delaunay polytope P is called strongly perfect ifits vertex set form a 4-design.

Theorem: A strongly perfect Delaunay polytope P is perfect.

Proof: If f(v) = 0 for v a vertex of P then

1

n

S

n1

f(x)2dx =1

| vert P| vvertP

f(v)2 = 0


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II. Covering maxima

and characterization


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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.

The only general method for computing (L) is to compute

all Delaunay polytopes ofL

.


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Covering maxima and pessima

Definition: If f Erdahl>0(n) then we define theinhomogeneous Hermite minimum by

H(f) = f(det Qf)1/n

We then have if Q is a quadratic form for L

(L)2 = max{f : f Erdahl(n) and Qf = Q}

(L) = n max{H(f)n/2 : f Erdahl(n) with Qf = Q}

Definition: A lattice L is said to be a covering maximum if forevery lattice L, which is near L we have (L) > (L).

Definition: A lattice L is said to be a covering pessimum if for

every lattice L, which is near L we have (L) > (L) exceptin a set of measure 0.

Theorem: We have equivalence between: L is a covering maximum (resp. pessimum). For every f Erdahl(n) with f = Q and (L)

2 = f the

function H(f) is a covering maximum (resp. pessimum)


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Convexity results

Theorem: The function f f is convex on Erdahl(n). Proof: Write f as

f(x) = Qf[x cf] f

and we havef = min

xRnf(x)

which implies the result.

Theorem (Minkovski): The function f

df

= (det Qf

)1/n

is convex on Erdahl(n).

But the product f H(f) = fdf is not convex.


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Characterization of covering maxima

Theorem: Take f Erdahl>0(n). The following areequivalent:

f is a covering maximum for H. f is perfect and eutactic.

Proof: Suppose that f is perfect and eutactic. We have thedifferentiation formula:

H(f+ f) = H(f) 1(det Qf)1/n

uf, f + o(f)

We also have

uf, f =

vZ(f)v(f)(v)

with v > 0. Iff is not collinear to f then there existsv Z(f) with (f)(v) > 0. Thus H decreases in direction f.

The reverse implication follows by preceding convexity results.


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Characterization of covering pessima

Theorem: Take f Erdahl>0(n). The following areequivalent:

f is a covering pessimum for H f is weakly eutactic

Proof: Suppose that f is weakly eutactic. We have thedifferentiation formula:

H(f+ f) = H(f) 1(det Qf)1/n

uf, f + o(f)

We also have

uf, f =

vZ(f)v(f)(v)

with v 0. Iff is not collinear to f and chosen outside ofa set of measure 0 then (f)(v) > 0 for all v and uf, f > 0.

The reverse implication follows by preceding convexity results.

T l l M f


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Topological Morse function property

Call Symm(n) the space of n n symmetric matrices.

Theorem: Take a lattice L and assume that all the Delaunaypolytopes of maximal circumradius are eutactic. The followingare equivalent: L is a non-degenerate topological Morse point for There exist a proper subspace SPL of Symm(n) such that for

every f0 Erdahl(n) with Qf0 = Q0 and (L)2

= f0 we haveQ Symm(n) : f Erdahl(n),

Qf = Q and Z(f) = Z(f0)

SPL

Theorem: The covering density function Q (Q) is a

topological Morse function if and only if n 3.

Proof: The lattice D4 has three translation classes ofDelaunay polytopes with the subspaces having zerointersection and their sum spanning Symm(4).


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III. Examples

K f D l l


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Known perfect Delaunay polytopes

In dimension 6, there is only Gossets 221, which defines theDelaunay polytopes of E6.

No classification in higher dimension. For n = 7, conjectured only two cases:

Lattice # vertices # facets | Sym |E7 56 702 2903040 (Gossets 321)

ER7 35 228 1440 For n = 8, a conjectured list of 27 possibilities. For n = 9, at least 100000 perfect Delaunay polytopes.

Some infinite series were built in M. Dutour, Infinite serie of extreme Delaunay polytopes,

European J. Combin. 26 (2005) 129132. V.P. Grishukhin, Infinite series of extreme Delaunay polytopes,

European J. Combin. 27 (2006) 481495. R. Erdahl and K. Rybnikov, An infinite series of perfect

quadratic forms and big Delaunay simplices in Zn, Proc.Steklov Inst. Math. 239 (2002) 159167.

E l f i i d i i


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Example of covering maxima and minima

The following lattices are covering maxima:name design strength # vertices # orbits Delaunay polytopes

E6 4 27 1

E7 5 56 2ER7 0 35 4O10 3 160 6

BW16 5 512 4O23 7 94208 523 7 47104 709

The following lattices are covering pessima:name design strength # vertices # orbits Delaunay polytopesZn 3 2n 1Dn 3 2n1 2 (or 1)E6 2 9 1E7 3 16 1E8 3 16 2

K12 3 81 4 Proof method: Take L a lattice

Enumerate all Delaunay polytopes of L up to symmetry Select the ones of maximum circumradius. Check for perfection (linear system problem)

check for eutacticity, resp. weakly eutacticity (linearprogramming problem)

I fi it f i i


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Infinite sequence of covering maxima

We need to find explicit lattice.

If n even, n 6, there is a n-dimensional extreme Delaunay

EDn formed with 3 layers of Dn1 lattice a vertex the n 1 half-cube the n 1 cross-polytope

If n odd, n 7, there is a n-dimensional extreme Delaunay

EDn formed with 3 layers of Dn1 lattice the n 1 cross polytope the n 1 half-cube the n 1 cross polytope

ED6 is Gossets 221 and ED7 is Gossets 321.

For a Delaunay polytope P, we define L(P) to be the smallestlattice containing P.

Theorem: The lattice L(EDn) spanned by EDn is a coveringmaxima.

We need the full list of orbits of Delaunay of L(EDn).

Proof method


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Proof method

Suppose that we have p orbits O1, . . . ,Op of Delaunaypolytopes P1, . . . ,Pp then

1

pi=1

|Oi| vol Pi.

If equality then the list O1, . . . ,Op is complete.

Take P is a polytope c P a point invariant under Aut P. IfF1, . . . , Fr are r orbits Orb1, . . . ,Orbr of facets of P then wehave

vol P r

i=1

|Orbi| vol(conv(Fi, c))

If equality then the list F1, . . . Fr is complete.

We get a conjectured list of orbits of Delaunay polytope ofL(EDn), of facets of EDn and prove their completude by the

volume formula.

Non generating perfect Delaunay polytopes


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Non generating perfect Delaunay polytopes

Theorem: There exist perfect Delaunay polytopes P in lattices

L with L(P) = L. Proof:

Take 2 s and 4s n and define the n-polytope

J(n, s) = {x {0, 1}n+1 such thatn+1

i=1

xi = s}

Define the vector vn,s =

14

4s, 0n+14s

.

P(n, s) = J(n, s) 2vn,s J(n, s) is a perfect n-dimensionalDelaunay polytope in Ln,s = L(P(n, s)).

Fact: If6s

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Mathieu Dutour Sikiric, Konstantin Rybnikov, Achill Schuermann and Frank Vallentin- Covering maxima and Delaunay polytopes in lattices