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Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Parallelohedra and the Voronoi Conjecture

Alexey Garb er
Moscow State University and Delone Lab oratory of Yaroslavl State University, Russia

FU Berlin Novemb er 28, 2013

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Parallelohedra

Denition Convex

d

-dimensional p olytop e

P

is called a parallelohedron if

R

d

can b e (face-to-face) tiled into parallel copies of

P

.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Parallelohedra

Denition Convex

d

-dimensional p olytop e

P

is called a parallelohedron if

R

d

can b e (face-to-face) tiled into parallel copies of

P

.

Two typ es of two-dimensional parallelohedra

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Three-dimensional parallelohedra

In 1885 Russian crystallographer E.Fedorov listed all typ es of three-dimensional parallelohedra.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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Three-dimensional parallelohedra

In 1885 Russian crystallographer E.Fedorov listed all typ es of three-dimensional parallelohedra.

Parallelepip ed and hexagonal prism with centrally symmetric base.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Three-dimensional parallelohedra

In 1885 Russian crystallographer E.Fedorov listed all typ es of three-dimensional parallelohedra.

Rhombic do decahedron, elongated do decahedron, and truncated o ctahedron

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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Tiling by rhombic do decahedra

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Minkowski-Venkov conditions

Theorem (H.Minkowski, 1897, and B.Venkov, 1954)

P

is a

d

-dimensional parallelohedron i it satises the following

conditions:
1 2 3

P

is centrally symmetric;

Any facet of Projection of

P P

is centrally symmetric; along any its

(d - 2)-dimensional

face is

parallelogram or centrally symmetric hexagon.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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Minkowski-Venkov conditions

Theorem (H.Minkowski, 1897, and B.Venkov, 1954)

P

is a

d

-dimensional parallelohedron i it satises the following

conditions:
1 2 3

P

is centrally symmetric;

Any facet of Projection of

P P

is centrally symmetric; along any its

(d - 2)-dimensional

face is

parallelogram or centrally symmetric hexagon.

Theorem (N.Dolbilin and A.Magazinov, 2013) If

P

tiles

d

-dimensional space with p ositive homothetic copies

separated from 0 then

P

satises Minkowski-Venkov conditions.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Belts of parallelohedra
Denition The set of facets parallel to a given hexagon.

(d - 2)-face

is called b elt.

These facets are projected onto sides of a parallelogram or a

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Belts of parallelohedra
Denition The set of facets parallel to a given hexagon. There are 4-b elts

(d - 2)-face

is called b elt.

These facets are projected onto sides of a parallelogram or a

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Belts of parallelohedra
Denition The set of facets parallel to a given

(d - 2)-face

is called b elt.

These facets are projected onto sides of a parallelogram or a hexagon. There are 4-b elts and 6-b elts.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Parallelohedra and Lattices

Let

T

P b e the unique face-to-face tiling of

P.

Then centers of tiles forms a lattice

Rd P .

into parallel copies of

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Parallelohedra and Lattices I I
Consider we have an arbitrary arbitrary p oint

d

-dimensional lattice



and

O

of

.

O

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Parallelohedra and Lattices I I
Consider we have an arbitrary arbitrary p oint

d

-dimensional lattice



and

O

of

. O of
than ).

Consider a p olytop e consist of p oints that are closer to to any other lattice p oint (Dirichlet-Voronoi p olytop e

O

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Parallelohedra and Lattices I I
Consider we have an arbitrary arbitrary p oint

d

-dimensional lattice



and

O

of

. O of
than ).

Consider a p olytop e consist of p oints that are closer to to any other lattice p oint (Dirichlet-Voronoi p olytop e Then

DV

is a parallelohedron and p oints of



are centers of

corresp ondent tiles.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Voronoi conjecture

Conjecture (G.Voronoi, 1909) Every parallelohedron is ane equivalent to Dirichlet-Voronoi p olytop e of some lattice

.

-

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Some known results

Denition A

d
.

-dimensional parallelohedron

P

is called primitive if every

vertex of the corresp onding tiling b elongs to exactly

d+

1 copies of

P

Theorem (G.Voronoi, 1909) The Voronoi conjecture is true for primitive parallelohedra.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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Some known results

Denition A

d

-dimensional parallelohedron

P

is called

k

-primitive if every

k

-face of the corresp onding tiling b elongs to exactly

d +1-k

copies of

P

.

Theorem (O.Zhitomirskii, 1929) The Voronoi conjecture is true for b elts of length 4.

(d - 2)-primitive d

-dimensional

parallelohedra. Or the same, it is true for parallelohedra without

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Dual cells
Denition The dual cell of a face

F

of given parallelohedral tiling is the set of

all centers of parallelohedra that shares

F

. If

F

is

(d - k )-dimensional

then the corresp onding cell is called

k

-cell.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Dual cells
Denition The dual cell of a face

F

of given parallelohedral tiling is the set of

all centers of parallelohedra that shares

F

. If

F

is

(d - k )-dimensional

then the corresp onding cell is called

k

-cell.

The set of all dual cells of the tiling with corresp onding incidence relation determines a structure of a cell complex.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Dual cells
Denition The dual cell of a face

F

of given parallelohedral tiling is the set of

all centers of parallelohedra that shares

F

. If

F

is

(d - k )-dimensional

then the corresp onding cell is called

k

-cell.

The set of all dual cells of the tiling with corresp onding incidence relation determines a structure of a cell complex. Conjecture (Dimension conjecture) The dimension of dual

k

-cell is equal to

k

.

The dimension conjecture is necessary for the Voronoi conjecture.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Dual cells
Denition The dual cell of a face

F

of given parallelohedral tiling is the set of

all centers of parallelohedra that shares

F

. If

F

is

(d - k )-dimensional

then the corresp onding cell is called

k

-cell.

The set of all dual cells of the tiling with corresp onding incidence relation determines a structure of a cell complex. Conjecture (Dimension conjecture) The dimension of dual

k

-cell is equal to

k

.

The dimension conjecture is necessary for the Voronoi conjecture. Theorem (A.Magazinov, 2013) Dual
A.Garb er Parallelohedra and the Voronoi Conjecture

k

-cell has at most 2

k vertices.
MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Dual 3-cells and 4-dimensional parallelohedra
Lemma (B.Delone, 1929) There are ve typ es of three-dimensional dual cells: tetrahedron, o ctahedron, quadrangular pyramid, triangular prism and cub e. Theorem (B.Delone, 1929) The Voronoi conjecture is true for four-dimensional parallelohedra. Delone used this result to nd full classication of four-dimensional parallelohedra. He found 51 of them and the last 52nd was added by M.Shtogrin in 1973.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Dual 3-cells and 4-dimensional parallelohedra
Lemma (B.Delone, 1929) There are ve typ es of three-dimensional dual cells: tetrahedron, o ctahedron, quadrangular pyramid, triangular prism and cub e. Theorem (B.Delone, 1929) The Voronoi conjecture is true for four-dimensional parallelohedra. Delone used this result to nd full classication of four-dimensional parallelohedra. He found 51 of them and the last 52nd was added by M.Shtogrin in 1973. Theorem (A.Ordine, 2005) The Voronoi conjecture is true for parallelohedra without cubical or prismatic dual 3-cells.
A.Garb er Parallelohedra and the Voronoi Conjecture MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Equivalent Statement
Problem (Dual conjecture) For every parallelohedral tiling p ositive denite quadratic Dirichlet-Voronoi p olytop e

TP with lattice there exist a T form Q (x) = x Q x such that P is of with resp ect to metric dened

by

Q.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Equivalent Statement
Problem (Dual conjecture) For every parallelohedral tiling p ositive denite quadratic Dirichlet-Voronoi p olytop e Consider the dual tiling

TP with lattice there exist a T form Q (x) = x Q x such that P is of with resp ect to metric dened

by

Q.

TP .

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Equivalent Statement
Problem (Dual conjecture) For every parallelohedral tiling p ositive denite quadratic Dirichlet-Voronoi p olytop e Consider the dual tiling

TP with lattice there exist a T form Q (x) = x Q x such that P is of with resp ect to metric dened
tiling after appropriate ane

by

Q.

TP .This

transformation must b e the Delone tiling of image of lattice

.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Equivalent Statement
Problem (Dual conjecture) For every parallelohedral tiling p ositive denite quadratic Dirichlet-Voronoi p olytop e Consider the dual tiling

TP with lattice there exist a T form Q (x) = x Q x such that P is of with resp ect to metric dened
tiling after appropriate ane

by

Q.

TP .This

transformation must b e the Delone tiling of image of lattice Problem Prove that for dual tiling

.

TP there exist a positive denite quadratic form Q (x) = Q x (or an ellipsoid E that represents a unit sphere with resp ect to Q ) such that TP is a Delone tiling with resp ect to Q and centers of corresp onding empty ellipsoids are in vertices of tiling TP

xT

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Equivalent Statement I I

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Equivalent Statement I I

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Equivalent Statement I I

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Equivalent Statement I I

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Equivalent Statement I I

This approach was used by R.Erdahl in 1999 to prove the Voronoi conjecture for zonotop es.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Equivalent Statement I I

This approach was used by R.Erdahl in 1999 to prove the Voronoi conjecture for zonotop es. Several more equivalent reformulations can b e found in work of M.Deza and V.Grishukhin Prop erties of parallelotop es equivalent to Voronoi's conjecture, 2004.
A.Garb er Parallelohedra and the Voronoi Conjecture MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Canonical scaling

Denition A (p ositive) real-valued function conditions for facets

n(F )

dened on set of all facets of

tiling is called canonical scaling if it satises the following

F

i that contains arbitrary

(d - 2)-face G

:

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Canonical scaling

Denition A (p ositive) real-valued function conditions for facets

n(F )

dened on set of all facets of

tiling is called canonical scaling if it satises the following

F

i that contains arbitrary

(d - 2)-face G
2
2

:

F F
e
2

2

e e
1

e

1

G
e

F
3

F
1

G
3

F

1

e

3

e

4

F

3

F

4

±n(Fi )ei =
A.Garb er Parallelohedra and the Voronoi Conjecture

0

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Constructing canonical scaling

How to construct a canonical scaling for a given tiling

TP

?

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Constructing canonical scaling

How to construct a canonical scaling for a given tiling If two facets

TP

?

F

1

and

F

2

of tiling has a common

from 6-b elt then value of canonical scaling on denes value on

(d - 2)-face F1 uniquely

F

2

and vice versa.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Constructing canonical scaling

How to construct a canonical scaling for a given tiling If two facets

TP

?

F

1

and

F

2

of tiling has a common

from 6-b elt then value of canonical scaling on denes value on If facets

(d - 2)-face F1 uniquely

F F

2

and vice versa. has a common

F

1

and

2

(d - 2) F
2

-face from 4-b elt

then the only condition is that if these facets are opp osite then values of canonical scaling on

F

1

and

are equal.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Constructing canonical scaling

How to construct a canonical scaling for a given tiling If two facets

TP

?

F

1

and

F

2

of tiling has a common

from 6-b elt then value of canonical scaling on denes value on If facets

(d - 2)-face F1 uniquely

F F

2

and vice versa. has a common

F

1

and

2

(d - 2) F F
2

-face from 4-b elt

then the only condition is that if these facets are opp osite then values of canonical scaling on If facets

F F

1

and

are equal.

F

1

and

F

2

are opp osite in one parallelohedron then
1

values of canonical scaling on

and

2

are equal.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Voronoi's Generatrissa

Consider we have a canonical scaling dened on tiling

TP

.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Voronoi's Generatrissa

We will construct a piecewise linear generatrissa function

G : Rd - R
A.Garb er

.
MSU and Delone Lab of YSU

Parallelohedra and the Voronoi Conjecture


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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Voronoi's Generatrissa

Step 1: Put

G

equal to 0 on one of tiles.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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Voronoi's Generatrissa

Step 2: When we pass through one facet of tiling the gradient of changes accordingly to canonical scaling.
A.Garb er Parallelohedra and the Voronoi Conjecture

G

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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Voronoi's Generatrissa

Step 2: Namely, if we pass a facet add vector
A.Garb er Parallelohedra and the Voronoi Conjecture

F

with normal vector e then we

n (F )

e to gradient.
MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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Voronoi's Generatrissa

We obtain a graph of generatrissa function

G

.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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Voronoi's Generatrissa I I

What do es this graph lo oks like?
A.Garb er Parallelohedra and the Voronoi Conjecture MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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Prop erties of Generatrissa

The graph of generatrissa parab oloid.

G

lo oks like piecewise linear

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Prop erties of Generatrissa

The graph of generatrissa parab oloid.

G

lo oks like piecewise linear

And actually there is a parab oloid denite quadratic form its shells.

y=

xT

Q

x for some p ositive

Q

tangent to generatrissa in centers of

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Prop erties of Generatrissa

The graph of generatrissa parab oloid.

G

lo oks like piecewise linear

And actually there is a parab oloid denite quadratic form its shells.

y=

xT

Q

x for some p ositive

Q

tangent to generatrissa in centers of

Moreover, if we consider an ane transformation parab oloid into parab oloid

A

of this

y=

xT x then tiling

T

P will

transform into Voronoi tiling for some lattice.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Prop erties of Generatrissa

The graph of generatrissa parab oloid.

G

lo oks like piecewise linear

And actually there is a parab oloid denite quadratic form its shells.

y=

xT

Q

x for some p ositive

Q

tangent to generatrissa in centers of

Moreover, if we consider an ane transformation parab oloid into parab oloid

A

of this

y=

xT x then tiling

T

P will

transform into Voronoi tiling for some lattice.

So to prove the Voronoi conjecture it is sucient to construct a canonical scaling on the tiling

TP .
MSU and Delone Lab of YSU

Works of Voronoi, Zhitomirskii and Ordine based on this approach.
A.Garb er Parallelohedra and the Voronoi Conjecture


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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Necessity of Generatrissa

Lemma Tangents to parab ola in p oints midp oint of

A

and

B

intersects in the

AB

.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Necessity of Generatrissa

Lemma Tangents to parab ola in p oints midp oint of

A

and

B

intersects in the

AB

.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Necessity of Generatrissa

Lemma Tangents to parab ola in p oints midp oint of

A

and

B

intersects in the

AB

.

This lemma leads to the usual way of constructing the Voronoi diagram for a given p oint set. We lift p oints onto parab oloid

y = xT x

in

R

d +1 .

Construct tangent hyp erplanes. Take the intersection of upp er-halfspaces. And project this p olyhedron back on the initial space.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Gain function instead of canonical scaling
We know how canonical scaling should change when we pass from one facet to neighb or facet across primitive

(d - 2)-face

of

F

.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Gain function instead of canonical scaling
We know how canonical scaling should change when we pass from one facet to neighb or facet across primitive Denition We will call the multiple of canonical scaling that we achieve by passing across

(d - 2)-face

of

F

.

F

the gain function

g

on

F

.

For any generic curve non-primitive

on (d - 2)-faces

surface of

P

that do not cross

we can dene the value

g ( )

.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Gain function instead of canonical scaling
We know how canonical scaling should change when we pass from one facet to neighb or facet across primitive Denition We will call the multiple of canonical scaling that we achieve by passing across

(d - 2)-face

of

F

.

F

the gain function

g

on

F

.

For any generic curve non-primitive Lemma

on (d - 2)-faces

surface of

P

that do not cross

we can dene the value

g ( )

.

The Voronoi conjecture is true for

P

i for any generic cycle

g ( ) =
A.Garb er

1.
MSU and Delone Lab of YSU

Parallelohedra and the Voronoi Conjecture


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Prop erties of gain function
Denition Consider a manifold manifold the

P

that is a surface of parallelohedron

P

with

deleted closed non-primitive

(d - 2)-faces.
.

We will call this



-surface of

P

The gain function is well dened on any cycle on

P

.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Prop erties of gain function
Denition Consider a manifold manifold the

P

that is a surface of parallelohedron

P

with

deleted closed non-primitive

(d - 2)-faces.
.

We will call this



-surface of

P

The gain function is well dened on any cycle on Lemma (A.Gavrilyuk, A.G., A.Magazinov) The gain function gives us a homomorphism

P

.

g : 1 (P ) - R
and the Voronoi conjecture is true for trivial.
A.Garb er Parallelohedra and the Voronoi Conjecture

+

P

i this homomorphism is

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Improvement
It is easy to see that values of canonical scaling should b e equal on opp osite facets of of

P

. So we can consider a



-surface

P

that obtained from

P

by gluing its opp osite p oints.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Improvement
It is easy to see that values of canonical scaling should b e equal on opp osite facets of of

P

. So we can consider a



-surface
that

P

that obtained from

P

by gluing its opp osite p oints.

We already know some cycles (half-b elt cycles) on

P

g

maps into 1. For example, any cycle formed by three facets

F1 , F2 , F3 that are face G (like three

parallel to primitive

(d - 2)

-dimensional

consecutive sides of a hexagon).

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Improvement
It is easy to see that values of canonical scaling should b e equal on opp osite facets of of

P

. So we can consider a



-surface
that

P

that obtained from

P

by gluing its opp osite p oints.

We already know some cycles (half-b elt cycles) on

P

g

maps into 1. For example, any cycle formed by three facets

F1 , F2 , F3 that are parallel to primitive (d - 2)-dimensional face G (like three consecutive sides of a hexagon). The group R+ is commutative so image of commutator subgroup [1 (P )] is trivial. Therefore, we factorize by
commutator and get the group of one-dimensional homologies over

Z

instead of fundamental group.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Improvement
It is easy to see that values of canonical scaling should b e equal on opp osite facets of of

P

. So we can consider a



-surface
that

P

that obtained from

P

by gluing its opp osite p oints.

We already know some cycles (half-b elt cycles) on

P

g

maps into 1. For example, any cycle formed by three facets

F1 , F2 , F3 that are parallel to primitive (d - 2)-dimensional face G (like three consecutive sides of a hexagon). The group R+ is commutative so image of commutator subgroup [1 (P )] is trivial. Therefore, we factorize by
commutator and get the group of one-dimensional homologies over

Z

instead of fundamental group. since there is no torsion in the group .

Moreover we can exclude the torsion part of the group

H1 (P , Z)
A.Garb er

R+

MSU and Delone Lab of YSU

Parallelohedra and the Voronoi Conjecture


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Improvement
It is easy to see that values of canonical scaling should b e equal on opp osite facets of of

P

. So we can consider a



-surface
that

P

that obtained from

P

by gluing its opp osite p oints.

We already know some cycles (half-b elt cycles) on

P

g

maps into 1. For example, any cycle formed by three facets

F1 , F2 , F3 that are parallel to primitive (d - 2)-dimensional face G (like three consecutive sides of a hexagon). The group R+ is commutative so image of commutator subgroup [1 (P )] is trivial. Therefore, we factorize by
commutator and get the group of one-dimensional homologies over

Z

instead of fundamental group. since there is no torsion in the group .

Moreover we can exclude the torsion part of the group

H1 (P , Z)
A.Garb er

R+

Finally we get the group
Parallelohedra and the Voronoi Conjecture

H1 (P , Q).
MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

The new result on Voronoi conjecture

Theorem (A.Gavrilyuk, A.G., A.Magazinov) The Voronoi conjecture is true for parallelohedra with trivial group

1 (P )
In

, i.e. for p olytop es with simply connected



-surface.

R

3

: cub e, rhombic do decahedron and truncated o ctahedron.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

The new result on Voronoi conjecture

Theorem (A.Gavrilyuk, A.G., A.Magazinov) The Voronoi conjecture is true for parallelohedra with trivial group

1 (P )
In

, i.e. for p olytop es with simply connected



-surface.

R

3

: cub e, rhombic do decahedron and truncated o ctahedron.

After applying all improvements we get:

Theorem (A.Gavrilyuk, A.G., A.Magazinov) If group of one-dimensional homologies of parallelohedron

H1 (P , Q)

of the



-surface

P

is generated by half-b elt cycles then the

Voronoi conjecture is true for

P

.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

How one can apply this theorem?

We start from a parallelohedron

P

.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

How one can apply this theorem?

Then put a vertex of graph

G

for every pair of opp osite facets.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

How one can apply this theorem?

Draw edges of

G

b etween pairs of facets with common primitive

(d - 2)-face.
A.Garb er Parallelohedra and the Voronoi Conjecture MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

How one can apply this theorem?

List all basic cycles half-b elt cycles.



that has gain function 1 for sure. These are

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

How one can apply this theorem?

List all basic cycles



that has gain function 1 for sure. These are

half-b elt cycles. And trivially contractible cycles around

(d - 3)-face.
A.Garb er Parallelohedra and the Voronoi Conjecture MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

How one can apply this theorem?

Check that basic cycles generates all cycles of graph

G

.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Three- and four-dimensional parallelohedra

Using describ ed algorithm we can check that every parallelohedron in

R

3

and

R

4

has homology group

H1 (P , Q)

generated by

half-b elts cycles and therefore it satises our condition.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Uniqueness theorem
Assume that Voronoi conjecture is true for parallelohedron

P

. How

many there are Voronoi p olytop es that are anely equivalent to

P

?

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Uniqueness theorem
Assume that Voronoi conjecture is true for parallelohedron

P

. How

many there are Voronoi p olytop es that are anely equivalent to Theorem (L.Michel, S.Ryshkov, M.Senechal, 1995) If

P

?

P
.

is primitive then there is unique Voronoi p olytop e equivalent to

P

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Uniqueness theorem
Assume that Voronoi conjecture is true for parallelohedron

P

. How

many there are Voronoi p olytop es that are anely equivalent to Theorem (L.Michel, S.Ryshkov, M.Senechal, 1995) If

P

?

P
.

is primitive then there is unique Voronoi p olytop e equivalent to

P

Theorem (N.Dolbilin, J.-i.Itoh, C.Nara, 2011) If graph

G

of

P

is connected then there is at most one Voronoi

p olytop e equivalent to

P

.

Theorem (A.Gavrilyuk, to app ear) If

G

has

k

comp onents then the set of Voronoi p olytop es

equivalent to
A.Garb er

P

is either empty or a

k

-orbifold.
MSU and Delone Lab of YSU

Parallelohedra and the Voronoi Conjecture


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Extension of Parallelohedra

Denition A vector

v

is called free with resp ect to parallelohedron

P

if the

Minkowski sum

P + 2 [-v , v ]

1

is a parallelohedron.

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

Extension of Parallelohedra

Denition A vector

v

is called free with resp ect to parallelohedron

P

if the

Minkowski sum

P + 2 [-v , v ]

1

is a parallelohedron.

Theorem (V.Grishukhin, 2006, corrected pro of in 2013 by A.Magazinov) A vector

v

is free with resp ect to

P

i

v

is parallel to at least one

facet from every 6-b elt.

Theorem (A.Magazinov, preprint) If vector

v

is free with resp ect to Voronoi p olytop e

P

then the

Voronoi conjecture is true for
A.Garb er Parallelohedra and the Voronoi Conjecture

P + 2 [-v , v ].
MSU and Delone Lab of YSU

1


Intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

More topics

THANK YOU!

A.Garb er Parallelohedra and the Voronoi Conjecture

MSU and Delone Lab of YSU