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Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Parallelohedra and the Voronoi Conjecture

Alexey Garb er

Bielefeld University Novemb er 28, 2012

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Parallelohedra

Denition Convex

d

-dimensional p olytop e

P

is called a parallelohedron if

Rd

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

P

.

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Parallelohedra

Denition Convex

d

-dimensional p olytop e

P

is called a parallelohedron if

Rd

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Tiling by rhombic do decahedra

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Prop erties of parallelohedra
Theorem (H.Minkowski, 1897) Any

d

-dimensional parallelohedron

P

satises the following

conditions:

1

P

is centrally symmetric;

2 Any facet of 3 Projection of

P P

is centrally symmetric; along any its

(d - 2)-dimensional

face is

parallelogram or centrally symmetric hexagon. The set of facets projected onto sides of such p olygon is called a b elt.

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Prop erties of parallelohedra
Theorem (H.Minkowski, 1897) Any

d

-dimensional parallelohedron

P

satises the following

conditions:

1

P

is centrally symmetric;

2 Any facet of 3 Projection of

P P

is centrally symmetric; along any its

(d - 2)-dimensional

face is

parallelogram or centrally symmetric hexagon. The set of facets projected onto sides of such p olygon is called a b elt.

Theorem (B.Venkov, 1954) Minkowski conditions are parallelohedron.
A.Garb er Parallelohedra and the Voronoi Conjecture

sucient

for convex p olytop e

P

to b e a


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Known results

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

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

Theorem (R.Erdahl, 1999) The Voronoi conjecture is true for zonotop es.

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Dual cells

Denition The dual cell for 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 ondent cell is called

k

-cell.

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Dual cells

Denition The dual cell for 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 ondent cell is called

k

-cell.

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

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Dual cells

Denition The dual cell for 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 ondent cell is called

k

-cell.

The set of all dual cells of the tiling with corresp ondent 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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Dual 3-cells and 4-dimensional parallelohedra

There are ve combinatorial typ es of three-dimensional dual cells: tetrahedron, o ctahedron, quadrangular pyramid, triangular prism and cub e.

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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 Q such that P is form Q ( ) = of with resp ect to metric dened

xxx

by

Q.

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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 Q such that P is form Q ( ) = of with resp ect to metric dened

xxx

by

Q.

TP .

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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 Q such that P is form Q ( ) = of with resp ect to metric dened

xxx

by

Q.

TP .This

tiling after appropriate ane

transformation must b e the Delone tiling of image of lattice

.

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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 Q such that P is form Q ( ) = of with resp ect to metric dened

xxx

by

Q.

TP .This

tiling after appropriate ane

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 ( ) = Q (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 ondent empty ellipsoids are in vertices of tiling TP

xxx
T

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Equivalent Statement I I

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Equivalent Statement I I

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Equivalent Statement I I

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Equivalent Statement I I

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Equivalent Statement I I

This approach was used by Erdahl for zonotop es.

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Equivalent Statement I I

This approach was used by Erdahl for zonotop es. Several more equivalent reformulations can b e found in work of Deza and Grishukhin Prop erties of parallelotop es equivalent to Voronoi's conjecture, 2003.
A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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 )
A.Garb er Parallelohedra and the Voronoi Conjecture

ei = 0


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Constructing canonical scaling

How to construct a canonical scaling for a given tiling

TP

?

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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.

F

1 and

2 has a common

(d - 2) F

-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

2 are equal.

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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.

F

1 and

2 has a common

(d - 2) F F

-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 1 and

2 are equal. 2 are equal.

F

1 and

F

2 are opp osite in one parallelohedron then

values of canonical scaling on

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Voronoi's Generatrissa

Consider we have a canonical scaling dened on tiling

TP

.

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Voronoi's Generatrissa

We will construct a piecewise linear generatrissa function

G : Rd - R
A.Garb er

.

Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Voronoi's Generatrissa

Step 1: Put

G

as 0 on one of tiles.

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Voronoi's Generatrissa

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

n (F )

e

F

with normal vector

to gradient.

e

then we


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Voronoi's Generatrissa

We obtain a graph of generatrissa function

G

.

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Voronoi's Generatrissa I I

What do es this graph lo oks like?
A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

Prop erties of Generatrissa

The graph of generatrissa parab oloid.

G

lo oks like piecewise linear

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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=

Q

tangent to generatrissa in centers of

xT Q x

for some p ositive

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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=

Q

tangent to generatrissa in centers of

xT Q x

for some p ositive

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

y=

transform into Voronoi tiling for some lattice.

xx
T

A

of this

then tiling

T

P will

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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=

Q

tangent to generatrissa in centers of

xT Q x

for some p ositive

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

y=

transform into Voronoi tiling for some lattice.

xx
T

A

of this

then tiling

T

P will

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

TP .

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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)-facet F

.

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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)-facet 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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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)-facet 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.

Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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 group of one-dimensional homologies over instead of fundamental group.

R

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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 group of one-dimensional homologies over instead of fundamental group. Moreover we can exclude the torsion part of the group

R

H1 (P , R)
A.Garb er

since there is no torsion in the group

R

+.

Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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 group of one-dimensional homologies over instead of fundamental group. Moreover we can exclude the torsion part of the group

R

H1 (P , R)
A.Garb er

since there is no torsion in the group

R

+.

Finally we get the group
Parallelohedra and the Voronoi Conjecture

H1 (P , Q).


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

How one can apply this theorem?

We start from a parallelohedron

P

.

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

How one can apply this theorem?

Check that basic cycles generates all cycles of graph

G

.

A.Garb er Parallelohedra and the Voronoi Conjecture


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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


Short intro duction

Dual approach

Voronoi's metho d of canonical scaling

Gain function

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.

THANK YOU!

A.Garb er Parallelohedra and the Voronoi Conjecture