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Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Belt diameter of some class of space lling zonotopes
Alexey Garber
Moscow State University and Delone Lab oratory of Yaroslavl State University, Russia

Alexandro Readings, Moscow State University May 22, 2012

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Polytopes with centrally symmetric facets
Consider a family PC of all d -dimensional polytopes with centrally symmetric facets. Theorem (A.D. Alexandrov anf G.C. Shephard)
Every polytope from PC is centrally symmetric itself.

Let P be any polytope from PC .

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Polytopes with centrally symmetric facets
Consider a family PC of all d -dimensional polytopes with centrally symmetric facets. Theorem (A.D. Alexandrov anf G.C. Shephard)
Every polytope from PC is centrally symmetric itself.

Let P be any polytope from PC . Denition For any (d - 2)-dimensional face F of P we can dene a belt BP (F ) as a set of all facets of P parallel to F .

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Polytopes with centrally symmetric facets
Consider a family PC of all d -dimensional polytopes with centrally symmetric facets. Theorem (A.D. Alexandrov anf G.C. Shephard)
Every polytope from PC is centrally symmetric itself.

Let P be any polytope from PC . Denition For any (d - 2)-dimensional face F of P we can dene a belt BP (F ) as a set of all facets of P parallel to F . We can construct it by moving from one facet to another across (d - 2)-faces parallel to F .
A.Garb er Belt diameter of zonotop es MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Constructing belts

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Constructing belts

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Constructing belts

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Constructing belts

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Constructing belts

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Constructing belts

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Constructing belts

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Belt distance and belt diameter
Denition
Belt distance between two facets of P is the minimal number of belts that we need to travel from one facets to another.

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Belt distance and belt diameter
Denition
Belt distance between two facets of P is the minimal number of belts that we need to travel from one facets to another.

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Belt distance and belt diameter
Denition
Belt distance between two facets of P is the minimal number of belts that we need to travel from one facets to another.

Denition
Belt diameter of P is the maximal belt distance between its facets.
A.Garb er Belt diameter of zonotop es MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Why this is interesting?
Conjecture (G.Voronoi, 1909)
Every convex polytope that tiles space with translation copies is anely equivalent to Dirichlet-Voronoi polytope of some lattice.

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Why this is interesting?
Conjecture (G.Voronoi, 1909)
Every convex polytope that tiles space with translation copies is anely equivalent to Dirichlet-Voronoi polytope of some lattice.

One of the most popular (and one the most successful for now) approaches to prove the Voronoi conjecture in parallelohedra theory uses a canonical scaling function. And values of canonical scaling can be uniquely dened on facets in one belt of length equal to 6.

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Questions

Problem What is the maximal belt diameter of d -dimensional polytope?

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Questions

Problem What is the maximal belt diameter of d -dimensional polytope? It is easy to see that answer for d = 3 is 2.

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Questions

Problem What is the maximal belt diameter of d -dimensional polytope? It is easy to see that answer for d = 3 is 2. We can add some restriction for maximal length of any belt or we can restrict ourselves only to certain subfamily of polytopes with centrally symmetric facets. Or both.

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Small belt lengths
There is only one d -dimensional polytope with all belts consist of 4 facets. It is the d -dimensional cube and its belt diameter is equal to 1.

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Small belt lengths
There is only one d -dimensional polytope with all belts consist of 4 facets. It is the d -dimensional cube and its belt diameter is equal to 1. The situation is much more interesting for those polytopes who has belts of length 4 or 6. It was proved that conditions to have centrally symmetric facets and belts of length at most 6 are necessary (H.Minkowski, 1897) and sucient (B.Venkov, 1954) for convex polytope to be a parallelohedron, i.e. to tile Rd with parallel copies.

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Small belt lengths
There is only one d -dimensional polytope with all belts consist of 4 facets. It is the d -dimensional cube and its belt diameter is equal to 1. The situation is much more interesting for those polytopes who has belts of length 4 or 6. It was proved that conditions to have centrally symmetric facets and belts of length at most 6 are necessary (H.Minkowski, 1897) and sucient (B.Venkov, 1954) for convex polytope to be a parallelohedron, i.e. to tile Rd with parallel copies. There are 5 three-dimensional parallelohedra, 52 four-dimensional and dozens of thousands of ve-dimensional. And maximal belt diameter is unknown for all cases except R .
3
A.Garb er Belt diameter of zonotop es MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Zonotopes
Denition A polytope is called a zonotope if it is a porjection of cube. Or equivalently it is a Minkowski sum of nite number of segments. In that case we will denote this polytope as Z (U ) where U is the correspondent vector set.

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Zonotopes
Denition A polytope is called a zonotope if it is a porjection of cube. Or equivalently it is a Minkowski sum of nite number of segments. In that case we will denote this polytope as Z (U ) where U is the correspondent vector set. The equivalent property is the following. Theorem (P. McMullen, 1980)
For d > 3 a d -dimensional polytope is a zonotope if and only if all (d - 2)-faces of P are centrally symmetric.

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Questions about belt diameter of zonotopes
Problem What is the maximal belt diameter of d -dimensional space-lling zonotopes, i.e. for zonotopes with belt length at most 6?

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Questions about belt diameter of zonotopes
Problem What is the maximal belt diameter of d -dimensional space-lling zonotopes, i.e. for zonotopes with belt length at most 6? Problem What is the maximal belt diameter of d -dimensional zonotopes with belt length at most 2k , k > 3?

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Questions about belt diameter of zonotopes
Problem What is the maximal belt diameter of d -dimensional space-lling zonotopes, i.e. for zonotopes with belt length at most 6? Problem What is the maximal belt diameter of d -dimensional zonotopes with belt length at most 2k , k > 3? Claim
If k d or if we do not restrict he maximal value of belt length then the answer is d - 1 and it is sharp.
A.Garb er Belt diameter of zonotop es MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

An idea how to get an upper bound
Denition Two sets E and F of d - 1 vectors in Rd each are called conjugated if dim(E fi ) = dim(F ej = d . Correspondent zonotope Z (E F ) is called symmetric. Theorem (A.G.)
The maximal belt diameter for zonotope is also achieved on some symmetric zonotope of smaller or equal dimension.

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

A new basis for symmetric zonotopes

Lemma
For space-lling zonotopes we can nd an upper bound only for zonotopes with set of zone vectors
V= A Ed - 1 0...0 1...1 ,

where A is a 0/1-matrix with at least half of zeros in each row.

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Results on belt diameter of zonotopes
Theorem (A.G.)
Belt diameter of d -dimensional space-lling zonotope is not greater than

log

2

4 d 5

.

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

Results on belt diameter of zonotopes
Theorem (A.G.)
Belt diameter of d -dimensional space-lling zonotope is not greater than

log

2

4 d 5

.

Using similar technique for arbitrary belt length we can prove Claim
Belt diameter of d -dimensional zonotope with belt length at most k does not exceed

log

1

+

1

k -2

d

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

-zonotopes and their diameters

Denition If U is a subset of the vector set E (d ) = {ei - ej }d,+ where ei is i j= the standard basis in Rd + then zonotope Z (U ) is called a -zonotope.
1 1 1

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

-zonotopes and their diameters

Denition If U is a subset of the vector set E (d ) = {ei - ej }d,+ where ei is i j= the standard basis in Rd + then zonotope Z (U ) is called a -zonotope.
1 1 1

Theorem (A.G.)
Belt diameter of d -dimensional -zonotope is not greater than 2 if d 6 and 3 if d > 6. These bounds are sharp in any dimension.

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU


Centrally symmetric facets

Belt diameter

Zonotop es

Symmetric zonotop es

THANK YOU!

A.Garb er Belt diameter of zonotop es

MSU and Delone Lab of YSU