Mathieu Dutour Sikiric- Perfect forms and perfect Delaunay polytopes
Transcript of Mathieu Dutour Sikiric- Perfect forms and perfect Delaunay polytopes
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Perfect forms and perfect Delaunay polytopes
Mathieu Dutour SikiricRudjer Boskovic Institute, Croatia
November 25, 2011
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I. Lattices, packings
and coverings
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Lattice packings
A lattice L Rn is a set of the form L = Zv1 + + Zvn.
A packing is a family of balls Bn(xi, r), i I of the sameradius r and center xi such that their interiors are disjoint.
If L is a lattice, the lattice packing is the packing defined bytaking the maximal value of > 0 such that L + Bn(0, ) is apacking.
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Density of lattice packings
Take the lattice packing defined by a lattice L:
(L)
The packing density has the expression
(L) =
(L)n vol(Bn(0, 1))
det L with (L) =
1
2 minvL{0} ||v||,
vol(Bn(0, 1)) the volume of the unit ball Bn(0, 1) and det Lthe volume of an unit cell.
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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 Euclidean spaceRn
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Lattice covering
For a lattice L we define the covering radius (L) to be thesmallest r such that the family of balls v + B
n(0, r) for v
L
cover Rn.
The covering density has the expression
(L) =
(L)n vol(Bn(0, 1))
det(L) 1
with (L) being the largest radius of Delaunay polytopes
The only general method for computing (L) is to computeall Delaunay polytopes of L.
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II. Gram matrix
formalism
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Gram matrix and lattices
Denote by Sn the vector space of real symmetric n n
matrices andSn
>0 the convex cone of real symmetric positivedefinite n n matrices.
Take a basis (v1, . . . , vn) of a lattice L and associate to it theGram matrix Gv = (vi, vj)1i,jn Sn>0.
Example: take the hexagonal lattice generated by v1 = (1, 0)
and v2 =
12 ,
32
2v
v1
Gv =12
2 11 2
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Isometric lattices
Take a basis (v1, . . . , vn) of a lattice L withvi = (vi,1, . . . , vi,n) R
n and write the matrix
V =
v1,1 . . . vn,1...
. . ....
v1,n . . . vn,n
and Gv = VT V.
The matrix Gv is defined byn(n+1)
2 variables as opposed to n2
for the basis V.
If M Sn>0, then there exists V such that M = VT V (Gram
Schmidt orthonormalization) If M = VT1 V1 = V
T2 V2, then V1 = OV2 with O
T O = In(i.e. O corresponds to an isometry ofRn).
Also if L is a lattice ofRn with basis v and u an isometry of
R
n
, then Gv = Gu(v).
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Arithmetic minimum
The arithmetic minimum of A Sn>0 is
min(A) = minxZn{0}
xTAx
The minimal vector set of A Sn>0 is
Min(A) =
x Zn | xTAx = min(A)
Both min(A) and Min(A) can be computed using someprograms (for example sv by Vallentin)
The matrix Ahex =
2 11 2
has
Min(Ahex) = {(1, 0), (0, 1), (1, 1)}.
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Reexpression of previous definitions
Take a lattice L = Zv1 + + Zvn. If x L,
x = x1v1 + + xnvn with xi Z
we associate to it the column vector X =
x1...
xn
We get ||x||2 = XTGvX and
det L =
det Gv and (L) =1
2
min(Gv)
(L)
For Ahex = 2 11 2
, det Ahex = 3 and min(Ahex) = 2
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Changing basis
Ifv and v are two basis of a lattice L then V = VP with
P GLn(Z
). This implies
Gv = VTV = (VP)TVP = PT{VTV}P = PTGvP
If A, B Sn>0, they are called arithmetically equivalent if there
is at least one P GLn(Z
) such that
A = PTBP
Lattices up to isometric equivalence correspond to Sn>0 up to
arithmetic equivalence. In practice, Plesken/Souvignier wrote a program isom for
testing arithmetic equivalence and a program autom forcomputing automorphism group of lattices.All such programs take Gram matrices as input.
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An example of equivalence
Take the hexagonal lattice and two basis in it.
v1
v2
v1 = (1, 0) and v2 =
12 ,
3
2
2v
v1
v1 =
52 ,
3
2
and v2 = (1, 0)
One has v1 = 2v1 + v2, v2 = v1 and P = 2 1
1 0
Gv =
1 1212 1
and Gv =
7 52
52 1
= PTGvP
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Root lattices Let us take the lattice
An =
x Zn+1
s.t.
n+1i=1
xi = 0
If we take the basis vi = ei+1 ei then we get the Grammatrix A = (aij)1i,jn with ai,i = 2, ai,i+1 = ai+1,i = 1 andai,j = 0 otherwise.
Let us take the lattice
Dn =
x Zn s.t.
ni=1
xi 0 (mod 2)
For the basis v1 = e1 + e2, v2 = e1 e2, vi = ei ei
1 we get
Gv =
2 0 1 0 . . . 00 2 1 0 . . . 01 1 2 1 . . . 0
0 0 1 2. . . 0
..
. 0
. . .. . .
. . .10 . . . . . . . . . 1 2
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III. Perfect and
eutactic forms
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Hermite function
If A Sn>0 then the arithmetic minimum is
min(A) = minxZn{0} xT
Ax
and the set of minimal vectors is
Min(A) = x Zn : xTAx = min(A)
The Hermite function on the space Sn>0 is
(A) =min(A)
(det A)1/n
The density of the lattice packing L associated to A is
(L) =
(A)nvol(Bn(0, 1))
2n
Finding lattice packings with highest packing density is the
same as maximizing the Hermite function.
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Perfect forms
A form A is extreme if there is a neighborhood V of A in Sn>0
such that
If B V with B = A then (B) < (A)
A matrix A Sn>0 is perfect (Korkine & Zolotarev, 1873) if
the equation
B Sn and xTBx = min(A) for all x Min(A)
implies B = A.
Theorem: (Korkine & Zolotarev, 1873) If a form is extremethen it is perfect.
Perfect forms are rational forms.
If A is perfect then (A)n is rational.
f f
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A perfect form
Ahex = 2 1
1 2
corresponds to the lattice:
v1
v2
If B =
a b
b c
satisfies to xTBx = min(Ahex) for
x Min(Ahex) = {(1, 0), (0, 1), (1, 1)}, then:
a = 2, b = 2 and a 2c + b = 2
which implies B = Ahex. Ahex is perfect.
A f f
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A non-perfect form
Asqr =
1 00 1
has Min(Asqr) = {(0, 1), (1, 0)}.
See below lattices LB, Lsqr associated to matricesB, Asqr S
2>0 with Min(B) = Min(Asqr):
v1
v2
v1
v2
E i f
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Eutactic forms
A form A Sn>0
is eutactic (Voronoi, 1908) if there existv > 0 such that
A1 =
vMin(A)vvv
T
Theorem: (Voronoi, 1908) A form A is extreme if and only ifit is perfect and eutactic.
Theorem: (Ash, 1977)
(i) If A is not an eutactic form then it is topologically ordinary
point for (ii) If A is an eutactic form then it is a critical but topologicallynon-degenerate point for .
(ii) is a topological Morse function.
E l f f t f
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Examples of perfect forms
The root lattice are all perfect:
Name Min |Min| det | Aut |An ei ej 2n(n + 1) n + 1 2(n + 1)!Dn ei ej 4n(n 1) 4 2
nn!E6 complex 72 3 103680E7 complex 126 2 2903040
E8 complex 240 1 696729600 Another remarkable lattice is the Leech lattice of dimension
24. Every vector v has v2 4 and det Leech = 1. There are 196280 shortest vectors (maximal number in
dimension 24) Its automorphism group quotiented by Id24 is the sporadic
simple group Co0 It plays a significant role in modular form theory and
Lorentzian lattice theory.
K lt l tti ki d it i i ti
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Known results on lattice packing density maximization
dim. Nr. of perfect forms Absolute maximumof realized by
2 1 (Lagrange) A23 1 (Gauss) A34 2 (Korkine & Zolotarev) D45 3 (Korkine & Zolotarev) D56 7 (Barnes) E6 (Blichfeldt)
7 33 (Jaquet) E7 (Blichfeldt)8 10916 (DSV) E8 (Blichfeldt)9 500000 9?
24 ? Leech (Cohn & Kumar)
Remarks
The enumeration of perfect forms is done with the Voronoialgorithm.
The solution in dimension 24 was obtained by different
methods.
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IV. Ryshkov cone
and the Voronoi algorithm
The Ryshkov cone
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The Ryshkov cone
The Ryshkov cone Rn is defined as
Rn =
A Sn s.t. xTAx 1 for all x Zn {0}
The cone is invariant under the action of GLn(Z).
The cone is locally polyhedral, i.e. for a given A Rnx Zn s.t. xTAx = 1
is finite
Vertices of Rn correspond to perfect forms. For a form A we define the local cone
Loc(A) =
Q Sn s.t. xTQx 0 for x Min(A)
The Voronoi algorithm
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The Voronoi algorithm
Find a perfect form (say An), insert it to the list L as undone.
Iterate For every undone perfect form A in L, compute the local coneLoc(A) and then its extreme rays.
For every extreme ray r of Loc(A) realize the flipping, i.e.compute the adjacent perfect form A = A + r.
If A is not equivalent to a form in L, then we insert it into Las undone.
Finish when all perfect domains have been treated.
The subalgorithms are:
Find the extreme rays of the local cone Loc(A) (use cdd or lrsor any other program)
For any extreme ray r of Loc(A) find the adjacent perfectform A in the Ryshkov cone Rn
Test equivalence of perfect forms using autom
Flipping on an edge I
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Flipping on an edge I
Min(Ahex) = {(1, 0), (0, 1), (1, 1)}
with
Ahex =
1 1/2
1/2 1
and D =
0 1
1 0
v1
v2
Ahex
Flipping on an edge II
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Flipping on an edge II
Min(B) = {(1, 0), (0, 1)}
with
B =
1 1/4
1/4 1
= Ahex + D/4
v1
v2
A
B
hex
Flipping on an edge III
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Flipping on an edge III
Min(Asqr) = {(1, 0), (0, 1)}
with
Asqr =
1 00 1
= Ahex + D/2
v1
v2
Ahex
Asqr
Flipping on an edge IV
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Flipping on an edge IV
Min(Ahex) = {(1, 0), (0, 1), (1, 1)}
with
Ahex =
1 1/2
1/2 1
= Ahex + D
v1
v2
Ahex
Ahex
The Ryshkov cone R2
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The Ryshkov cone R2
+ (1,1)
+ (1,2)
+ (1,0)
+ (2,1)
+ (2,1)
+ (1,1)
+ (0,1)
+ (1,2)
1/2
1 1/2
1
1/21
1/2 1
3
3/2
1
3
3/2
3/2
1
3/2
Well rounded forms and retract
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Well rounded forms and retract
A form Q is said to be well rounded if it admits vectors v1,. . . , vn such that
(v1, . . . , vn) form a basis ofRn
v1, . . . , vn are shortest vectors. Q[v1] = = Q[vn].
Well rounded forms correspond to bounded faces of Rn.
Every form can be continuously deformed to a well roundedform and this defines a retracting homotopy of Rn onto apolyhedral complex WRn of dimension
n(n1)2 .
Every face of WRn has finite stabilizer, hence we can use it forcomputing the homology of GLn(Z) and other arithmeticgroups.
Actually, in term of dimension, we cannot do better: A. Pettet and J. Souto, Minimality of the well rounded retract,
Geometry and Topology, 12 (2008), 1543-1556.
References
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References
G. Voronoi, Nouvelles applications des parametres continues a
la theorie des formes quadratiques 1: Sur quelques proprietesdes formes quadratiques positives parfaites, J. Reine Angew.Math 133 (1908) 97178.
M. Dutour Sikiric, A. Schurmann and F. Vallentin,Classification of eight dimensional perfect forms, Electron.
Res. Announc. Amer. Math. Soc. A. Schurmann, Computational geometry of positive definite
quadratic forms, University Lecture Notes, AMS.
J. Martinet, Perfect lattices in Euclidean spaces, Springer,
2003. S.S. Ryshkov, E.P. Baranovski, Classical methods in the
theory of lattice packings, Russian Math. Surveys 34 (1979)168, translation of Uspekhi Mat. Nauk 34 (1979) 363.
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V. The lattice covering
problem
Empty sphere and Delaunay polytopes
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p y p y p y p
A sphere S(c, r) of radius r and center c in an n-dimensional
latticeL
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, whosevertex-set is L S(c, r).
c
r
Equalities and inequalities
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q q
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
Iso-Delaunay domains
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y
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
2v
2v
1v
are part of the same iso-Delaunay domain. An iso-Delaunay domain is the assignment of Delaunay
polytopes, so it is also the assignment of the Voronoi polytopeof the lattice.
Primitive iso-Delaunay 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 iso-Delaunay domain is of dimension
n(n+1)2 , they are called primitive
Equivalence and enumeration
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The group GLn(Z) acts on Sn>0 by arithmetic equivalence and
preserve the primitive iso-Delaunay domains.
Voronoi proved that after this action, there is a finite numberof primitive iso-Delaunay domains.
Bistellar flipping creates one iso-Delaunay from a giveniso-Delaunay domain and a facet of the domain. In dim. 2:
Enumerating primitive iso-Delaunay domains is doneclassicaly:
Find one primitive iso-Delaunay domain. Find the adjacent ones and reduce by arithmetic equivalence.
This is very similar to the Voronoi algorithm for perfect forms.
The partition ofS2>0 R3 I
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>0
If q(x, y) = ux2 + 2vxy + wy2 then q S2>0 if and only ifv
2 < uw and u > 0.
w
v
u
The partition ofS2>0 R3 II
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We cut by the plane u + w = 1 and get a circle representation.
u
v
w
The partition ofS2>0 R3 III
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Primitive iso-Delaunay domains in S2>0:
Optimization problem
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The lattice covering problem is to find a lattice covering ofminimal density.
Thm. Given an iso-Delaunay domain LT, there exist a uniquelattice, which minimize the covering density over LT.
The effective lattice is obtained by solving a semidefiniteprogramming problem, so no exact solution, but approximate
solutions available at any precision. The local maxima that are found are defined by algebraic
integers. See for more details
A. Schurmann and F. Vallentin, Computational approaches tolattice packing and covering problems, Discrete &Compututational Geometry 35 (2006) 73116.
A. Schurmann, Computational geometry of positive definitequadratic forms, University Lecture Notes, AMS.
Known results on covering density minimization
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dim. Best covering nr of iso-Delaunay
2 A2 (Kershner) 1 (Voronoi)3 A3 (Bambah) 1(Voronoi)4 A4 (Delone & Ryshkov) 3(Voronoi)5 A5 (Ryshkov & Baranovski) 222(Engel)6 L6 (conj. Vallentin)? ?7 L7 (conj. Schurmann & Vallentin)? ?
24 Leech (conj.)? ?
It turn out that the lattice of minimal covering density areunique for n 5
In general the best lattice coverings are expected to be
non-rational and with low symmetry.
But experimentations seemed to indicate that E6 is a localcovering maxima.
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VI. Quadratic functions
and the Erdahl cone
The Erdahl cone
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Denote by E2(n) the vector space of degree 2 polynomialfunctions on Rn. 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 byan infinity of inequalities.
The group acting on Erdahl(n) is AGLn(Z), i.e. the group ofaffine integral transformations
x b+ Px for b Zn and P GLn(Z)
Scalar product
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Definition: If f, g E2(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}
Relation with Delaunay polytope
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If D is a Delaunay polytope of a lattice L = Zv1 + + Zvnof empty sphere S(c, r) then we define the function
fD,v : Zn R
x = (x1, . . . , xn) n
i=1 xivi c2 r2
Clearly fD,v Erdahl>0(n).
On the other hand if f Erdahl(n) then there exists a latticeL of dimension k n, a Delaunay polytope D of L, a basisv of L and a function AGLn(Z) such that
f (x1, . . . , xn) = fD,v(x1, . . . , xk)
Thus the faces of Erdahl(n) correspond to the Delaunaypolytope of dimension at most n.
The perfection rank of a Delaunay polytope is the dimensionof the face it defines in Erdahl(n).
Perfect quadratic function
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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+2
2 1 that is if the perfection rank is 1. This is equivalent to say that f defines an extreme ray in
Erdahl(n).
A perfect quadratic function is rational.
Perfect Delaunay polytope
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A Delaunay polytope P is perfect if vert P = Z(f) with f a
perfect quadratic function. It has at least
n+2
2
1 vertices.
Known results:dim. perfect Delaunay authors
1 [0, 1] in Z
2 3 4 5 (Deza, Laurent & Grishukhin)6 221 in E6 (Deza & Dutour)
7 321 in E7and ER7 in L(ER7)
8 27 (Dutour Sikiric & Rybnikov)9 100000 (Dutour Sikiric)
Voronoi algorithm on the Erdahl cone
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From a given n-dimensional perfect Delaunay polytope Q of
form f we can define the local cone
Loc(f) = {g E2(n) s.t. g(x) 0 for x Z(f)}
The flipping algorithm finds the adjacent quadratic perfect
form g from a given perfect form f. The problem is Erdahl(n) is not locally polyhedral, i.e. the
rank of g can be lower than n.
The technique is to use a recursive algorithm for realizing the
enumeration. We start form [0, 1] Rn
1
and by subdivizionreach [0, 1]n (its local cone is the cut cone CUTn+1 occuringin combinatorial optimization).
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VII. Covering maxima, pessima
and their characterization
Eutacticity
If f E d hl ( ) h d fi d h h
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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 +
fn
Q1f [x]
Definition: f Erdahl>0(n) is eutactic if uf is in the relativeinterior of 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 = vvertPv(v cP)(v cP)
T.
Covering maxima
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A given lattice L is called a covering maxima if for any latticeL near L we have (L) < (L).
Theorem: The following are equivalent: L is a covering maxima Every Delaunay polytope of maximal circumradius is perfect
and eutactic.
The following are perfect Delaunay polytope:
name # vertices # orbits Delaunay polytopesE6 27 1E7 56 2
ER7 35 4O10 160 6
BW16 512 4O23 94208 5
23 47104 709
Theorem: For any n 6 there exist one lattice L(DSn) whichis a covering maxima.There is only one perfect Delaunay polytope P(DSn) ofmaximal radius in L(DSn).
The infinite series
( )
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For n even P(DSn) is defined as the lamination over Dn1 of one vertex the half cube 1
2Hn
1 the cross polytope CPn1
For n = 6, it is E6. For n odd as the lamination over Dn1 of
the cross polytope CPn1 the half cube
12 Hn1
the cross polytope CPn1
For n = 7, it is E7. Conjecture: The lattice DSn has the following properties:
L(DSn) has the maximum covering density among all covering
maxima Among all perfect Delaunay polytopes, P(DSn) has
maximum number of vertices
maximum volume
If true this would imply Minkovski conjecture.
Pessimum and Morse function property
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For a lattice L let us denote Dcrit(L) the space of direction dof deformation of L such that increases in the direction d.
Definition: A lattice L is said to be a covering pessimum if thespace Dcrit is of measures 0.
Theorem: If a lattice L has all its Delaunay polytopes ofmaximum circumradius are eutactic and are not simplices then
Q is a pessimum.name # vertices # orbits Delaunay polytopesZn 2n 1
D4 8 1Dn (n 5) 2n1 2
E6 9 1
E
7 16 1E8 16 2K12 81 4
Theorem: The covering density function Q (Q) is atopological Morse function if and only if n 3.
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THANK
YOU