The Finiteness Threshold Width of Lattice Polytopes
Jan Hofmann
FU Berlin – Berlin Mathematical School

January 18, 2017

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

All of this is joint work with

Paco
MónicaChristian
The finiteness threshold width of lattice polytopes
arXiv:1607.00798
Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Classifying lattice polytopes
Some known classifications:

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Classifying lattice polytopes
Some known classifications:reflexive polytopes up to dimension 4 [KS09]

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Classifying lattice polytopes
Some known classifications:reflexive polytopes up to dimension 4 [KS09]
smooth reflexive polytopes up to dimension 9 [Ob07,LP08]

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Classifying lattice polytopes
Some known classifications:reflexive polytopes up to dimension 4 [KS09]
smooth reflexive polytopes up to dimension 9 [Ob07,LP08]
canonical 3-polytopes [Kas10]

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Classifying lattice polytopes
Some known classifications:reflexive polytopes up to dimension 4 [KS09]
smooth reflexive polytopes up to dimension 9 [Ob07,LP08]
canonical 3-polytopes [Kas10]
hollow 3-polytopes [AWW11,AKW15]

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Classifying lattice polytopes
Some known classifications:reflexive polytopes up to dimension 4 [KS09]
smooth reflexive polytopes up to dimension 9 [Ob07,LP08]
canonical 3-polytopes [Kas10]
hollow 3-polytopes [AWW11,AKW15]

Definition: Equivalent polytopes
Two lattice polytopes P and P 0 are called unimodularly equivalent, if there
is a lattice-preserving affine isomorphism mapping them onto each other,i.e. P 0 = AP + b, with A ∈ GLd (Z), b ∈ Zd .

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Classifying lattice polytopes
Some known classifications:reflexive polytopes up to dimension 4 [KS09]
smooth reflexive polytopes up to dimension 9 [Ob07,LP08]
canonical 3-polytopes [Kas10]
hollow 3-polytopes [AWW11,AKW15]

Definition: Equivalent polytopes
Two lattice polytopes P and P 0 are called unimodularly equivalent, if there
is a lattice-preserving affine isomorphism mapping them onto each other,i.e. P 0 = AP + b, with A ∈ GLd (Z), b ∈ Zd .
Finitely many lattice polytopes := finitely many up to the above
equivalence relation

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Classifying lattice polytopes

Naive approach: Enumerate by dimension and size (|P ∩ Zd |) of P
d = 1 works great:|P ∩ Zd |
2
3
4
..
.

# different polytopes
1
1
1
..
.

Jan Hofmann (FU Berlin – BMS)

,
,
..
.

The Finiteness Threshold Width,

January 18, 2017

Classifying lattice polytopes
Naive approach: Enumerate by dimension and size (n := |P ∩ Zd |) of P
d = 2 also works:
n
3
4
5
..
.

# different polytopes
1
3
6
..
.

Jan Hofmann (FU Berlin – BMS),The Finiteness Threshold Width

January 18, 2017

Classifying lattice polytopes
Naive approach: Enumerate by dimension and size (n := |P ∩ Zd |) of P
d = 2 also works:
n
3
4
5
..
.

# different polytopes
1
3
6
..
.

Jan Hofmann (FU Berlin – BMS),The Finiteness Threshold Width,. . .

January 18, 2017

Classifying lattice polytopes
Naive approach: Enumerate by dimension and size (n := |P ∩ Zd |) of P
d = 2 also works:
n
3
4
5
..
.

# different polytopes
1
3
6
..
.

,

,. . .

Works thanks to:
Pick’s Theorem, vol(P) = 2i + b − 2

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Classifying lattice polytopes
Naive approach: Enumerate by dimension and size (n := |P ∩ Zd |) of P
d = 2 also works:
n
3
4
5
..
.

# different polytopes
1
3
6
..
.

,

,. . .

Works thanks to:
Pick’s Theorem, vol(P) = 2i + b − 2 =⇒ vol(P) ≤ 2n − 5

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Classifying lattice polytopes
Naive approach: Enumerate by dimension and size (n := |P ∩ Zd |) of P
d = 2 also works:
n
3
4
5
..
.

# different polytopes
1
3
6
..
.

,

,. . .

Works thanks to:
Pick’s Theorem, vol(P) = 2i + b − 2 =⇒ vol(P) ≤ 2n − 5Every full-dimensional polytope contains a standard simplex.

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Classifying lattice polytopes
Naive approach: Enumerate by dimension and size (n := |P ∩ Zd |) of P
d = 3 fails miserably:

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Classifying lattice polytopes
Naive approach: Enumerate by dimension and size (n := |P ∩ Zd |) of P
d = 3 fails miserably:

Reeve tetrahedra:
0 1 0 1
Tr := conv 0 0 1 1
0 0 0 rvol(Tr ) = r =⇒Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

n
4

#P
∞

January 18, 2017

Rescue in dimension 3
Theorem[BS16a]
For each size n all but finitely many lattice 3-polytopes have width 1.

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Rescue in dimension 3
Theorem[BS16a]
For each size n all but finitely many lattice 3-polytopes have width 1.

Definition: (lattice) width of a polytope
The width of a lattice polytope P with respect to a linear functional` ∈ (Rd )∗ is defined as
width` (P) := max |` · p − ` · q| ,p,q∈P

and the (lattice) width of P is the minimum such width` (P) where ` ranges
over non-zero integer functionals:
width(P) :=Jan Hofmann (FU Berlin – BMS)

min`∈(Zd )∗ \{0}width` (P).

The Finiteness Threshold Width

January 18, 2017

Width examples

width(P) = 1

width(P) = 1

width(P) = 2

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Width quiz

width(P) = ?

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Width quiz

width(P) = ?

For more complicated examples, use polymake($P->LATTICE_WIDTH).

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Width quiz

width(P) = 4

For more complicated examples, use polymake($P->LATTICE_WIDTH).

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Enumerating lattice 3-polytopes
Theorem[BS16a]
For each size n all but finitely many lattice 3-polytopes have width 1.

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Enumerating lattice 3-polytopes
Theorem[BS16a]
For each size n all but finitely many lattice 3-polytopes have width 1.
# vertices
size 5size 6size 7
size 8size 9size 10
size 11

4
9
36
103
193
282
478
619

5
0
40
296
1195
2853
5985
11432

6
−
0
97
1140
5920
18505
48103

7
−
−
0
147
2491
16384
64256

8
−
−
−
0
152
3575
28570

9
−
−
−
−
0
108
3425

10
−
−
−
−
−
0
59

total
9
76
496
2675
11698
45035
156464

Table: Lattice 3-polytopes of width larger than one and size ≤ 11, classified
according to their size and number of vertices.

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Enumerating lattice 3-polytopes
Theorem[BS16a]
For each size n all but finitely many lattice 3-polytopes have width 1.
# vertices
size 5size 6size 7
size 8size 9size 10
size 11

4
9
36
103
193
282
478
619

5
0
40
296
1195
2853
5985
11432

6
−
0
97
1140
5920
18505
48103

7
−
−
0
147
2491
16384
64256

8
−
−
−
0
152
3575
28570

9
−
−
−
−
0
108
3425

10
−
−
−
−
−
0
59

total
9
76
496
2675
11698
45035
156464

Table: Lattice 3-polytopes of width larger than one and size ≤ 11, classified
according to their size and number of vertices.

How about the general case?
Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The finiteness threshold width

Definition: Finiteness threshold width
For each d and each n ≥ d + 1, denote by w ∞ (d, n) ∈ N ∪ {∞} theminimal width W ≥ 0 such that there exist only finitely many latticed-polytopes of size n and width > W .
Let w ∞ (d) := supn∈N w ∞ (d, n).

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The finiteness threshold width

Definition: Finiteness threshold width
For each d and each n ≥ d + 1, denote by w ∞ (d, n) ∈ N ∪ {∞} theminimal width W ≥ 0 such that there exist only finitely many latticed-polytopes of size n and width > W .
Let w ∞ (d) := supn∈N w ∞ (d, n).
w ∞ (1) = w ∞ (2) = 0

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The finiteness threshold width

Definition: Finiteness threshold width
For each d and each n ≥ d + 1, denote by w ∞ (d, n) ∈ N ∪ {∞} theminimal width W ≥ 0 such that there exist only finitely many latticed-polytopes of size n and width > W .
Let w ∞ (d) := supn∈N w ∞ (d, n).
w ∞ (1) = w ∞ (2) = 0
w ∞ (3) = 1

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The finiteness threshold width

Theoremw ∞ (d) < ∞Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The finiteness threshold width

Theorem

Theoremw ∞ (d) < ∞w ∞ (4) = 2

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The finiteness threshold width

Theorem

Theoremw ∞ (d) < ∞w ∞ (4) = 2

wH (d − 2) ≤ w ∞ (d) ≤ wH (d − 1)

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The finiteness threshold width

Theorem

Theoremw ∞ (d) < ∞w ∞ (4) = 2

wH (d − 2) ≤ w ∞ (d) ≤ wH (d − 1)

Definitions: Hollow polytopes, empty polytopes, wH (d) & wE (d)We say a lattice polytope is hollow, if P ∩ Zd ⊆ δP. We say it is empty, if
P ∩ Zd = vert(P).
We denote by wH (d) and wE (d) the maximum widths of hollow and empty
lattice d-polytopes, respectively.

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The finiteness threshold width

Theorem

Theoremw ∞ (d) < ∞w ∞ (4) = 2

Corollary
There are only finitely many empty 4-simplices of width larger than two.

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The finiteness threshold width

Theorem

Theoremw ∞ (d) < ∞w ∞ (4) = 2

Corollary
There are only finitely many empty 4-simplices of width larger than two.
This Corollary was proclaimed in [BBBK11], but the proof has a gap.

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The finiteness threshold width

Theorem

Theoremw ∞ (d) < ∞w ∞ (4) = 2

Corollary
There are only finitely many empty 4-simplices of width larger than two.
This Corollary was proclaimed in [BBBK11], but the proof has a gap.
Full classification of those polytopes was computed recently by Óscar
Iglesias, verifying a conjecture by Christian Haase[H00]

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The finiteness threshold width – is finite

Theoremw ∞ (d) < ∞Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The finiteness threshold width – is finite

Theoremw ∞ (d) < ∞w ∞ (d) ≤ wH (d − 1)

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The finiteness threshold width – is finite

Theoremw ∞ (d) < ∞3w ∞ (d) ≤ wH (d − 1) ≤ O(d 2 ) by the flatness theorem [KL88]

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Known values

d wE (d − 1) wH (d − 2) w ∞ (d) wH (d − 1)
1
−
−
0
−
2
1
−
0
1
1
1
1
2
3
4
1
2
2
3
≥4
3
≥4
≥4
5

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Known values

d wE (d − 1) wH (d − 2) w ∞ (d) wH (d − 1)
1
−
−
0
−
2
1
−
0
1
1
1
1
2
3
4
1
2
2
3
≥4
3
≥4
≥4
5
In particular wH (2) = 2 ≤ w ∞ (4) ≤ 3 = wH (3).

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Known values

d wE (d − 1) wH (d − 2) w ∞ (d) wH (d − 1)
1
−
−
0
−
2
1
−
0
1
1
1
1
2
3
4
1
2
2
3
≥4
3
≥4
≥4
5
In particular wH (2) = 2 ≤ w ∞ (4) ≤ 3 = wH (3).

wH (2) = 2

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Known values

d wE (d − 1) wH (d − 2) w ∞ (d) wH (d − 1)
1
−
−
0
−
2
1
−
0
1
1
1
1
2
3
4
1
2
2
3
≥4
3
≥4
≥4
5
In particular wH (2) = 2 ≤ w ∞ (4) ≤ 3 = wH (3).

wH (2) = 2
wH (3) = 3 [AWW11,AKW15]

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

How do hollow polytopes enter the picture?
How do d-polytopes of size n look like?

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

How do hollow polytopes enter the picture?
How do d-polytopes of size n look like?
Non-hollow P have bounded volume. [He83]
Only finitely many P of bounded volume. [LZ91]

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

How do hollow polytopes enter the picture?
How do d-polytopes of size n look like?
Non-hollow P have bounded volume. [He83]
Only finitely many P of bounded volume. [LZ91]=⇒ All but f. m. d-polytopes of size n are hollow.

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

How do hollow polytopes enter the picture?
How do d-polytopes of size n look like?
Non-hollow P have bounded volume. [He83]
Only finitely many P of bounded volume. [LZ91]=⇒ All but f. m. d-polytopes of size n are hollow.
All but f.m. hollow d-polytopes project onto a hollow(d − 1)-polytope.[NZ11]

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

How do hollow polytopes enter the picture?
How do d-polytopes of size n look like?
Non-hollow P have bounded volume. [He83]
Only finitely many P of bounded volume. [LZ91]=⇒ All but f. m. d-polytopes of size n are hollow.
All but f.m. hollow d-polytopes project onto a hollow(d − 1)-polytope.[NZ11]

Lemma
Let d < n ∈ N. All but finitely many lattice d-polytopes of size bounded
by n are hollow. Furthermore, all but finitely many of the hollow
d-polytopes admit a projection to some hollow lattice (d − 1)-polytope.

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

How do hollow polytopes enter the picture?
How do d-polytopes of size n look like?
Non-hollow P have bounded volume. [He83]
Only finitely many P of bounded volume. [LZ91]=⇒ All but f. m. d-polytopes of size n are hollow.
All but f.m. hollow d-polytopes project onto a hollow(d − 1)-polytope.[NZ11]

Lemma
Let d < n ∈ N. All but finitely many lattice d-polytopes of size bounded
by n are hollow. Furthermore, all but finitely many of the hollow
d-polytopes admit a projection to some hollow lattice (d − 1)-polytope.
=⇒ w ∞ (d) ≤ wH (d − 1)
Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Lifts of bounded size
Lemma
Let d < n ∈ N. All but finitely many lattice d-polytopes of size bounded
by n are hollow. Furthermore, all but finitely many of the hollow
d-polytopes admit a projection to some hollow lattice (d − 1)-polytope.

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Lifts of bounded size
Lemma
Let d < n ∈ N. All but finitely many lattice d-polytopes of size bounded
by n are hollow. Furthermore, all but finitely many of the hollow
d-polytopes admit a projection to some hollow lattice (d − 1)-polytope.

Definition: Lift of a polytope
We say that a (lattice) polytope P is a lift of a (lattice) (d − 1)-polytope
Q if there is a (lattice) projection π with π(P) = Q. Without loss ofgenerality, we will typically assume π : Rd → Rd−1 to be the map thatforgets the last coordinate.
Two lifts π1 : P1 → Q and π2 : P2 → Q are equivalent if there is a
unimodular equivalence f : P1 → P2 with π2 ◦ f = π1 .

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Lifts of bounded size
Theorem
Let Q ⊂ Rd−1 be a lattice (d − 1)-polytope of width W . Then all lifts
P ⊂ Rd of Q have width ≤ W . All but finitely many of them have width= W.

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Lifts of bounded size
Theorem
Let Q ⊂ Rd−1 be a lattice (d − 1)-polytope of width W . Then all lifts
P ⊂ Rd of Q have width ≤ W . All but finitely many of them have width= W.
Look at (d − 1)-polytope P and findconditions for P to have finitely many lifts.

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Lifts of bounded size
Theorem
Let Q ⊂ Rd−1 be a lattice (d − 1)-polytope of width W . Then all lifts
P ⊂ Rd of Q have width ≤ W . All but finitely many of them have width= W.
Look at (d − 1)-polytope P and findconditions for P to have finitely many lifts.
conditions for P to have infinitely many lifts.

Theorem
For all d ≥ 3, w ∞ (d) equals the maximum width of a lattice(d − 1)-polytope Q that admits infinitely many lifts of bounded size.
Moreover, Q is hollow.

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The case d = 4

Theoremw ∞ (4) = 2

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The case d = 4

Theoremw ∞ (4) = 2

There are 12 maximal hollow lattice 3-polytopes [AWW11,AKW15]

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The case d = 4

Theoremw ∞ (4) = 2

There are 12 maximal hollow lattice 3-polytopes [AWW11,AKW15]
Compute their width with polymake ( our algorithm is in version 3.0. )Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The case d = 4

Theoremw ∞ (4) = 2

There are 12 maximal hollow lattice 3-polytopes [AWW11,AKW15]
Compute their width with polymake ( our algorithm is in version 3.0. )5 out of the 12 have width 3.

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The case d = 4

Theoremw ∞ (4) = 2

There are 12 maximal hollow lattice 3-polytopes [AWW11,AKW15]
Compute their width with polymake ( our algorithm is in version 3.0. )5 out of the 12 have width 3.
All their subpolytopes have width at most 2.

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The case d = 4
Theoremw ∞ (4) = 2
Figure 1: The Z3 -maximal integral lattice-free polytopes with lattice width two. For
further reference, the polytopes are labeled by a pair of indices (i, j), where i is thenumber of facets and j the lattice diameter (defined at the end of the introduction).

Figure:The
five (maximal)
hollow 3-polytopes
width
three
Figure2: TheZ3 -maximal
integral lattice-freepolytopes withoflattice
widththree.
Proof strategy
Jan Hofmann
Berlin– BMS) 1, weTheThresholdWidth). 2017
In the(FUproofof Theorem
useFinitenessa classificationof allZ2 -maximal polytopes inJanuaryP( 1 Zd18,

The case d = 4

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The case d = 4

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The case d = 4

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The case d = 4
Theoremw ∞ (4) = 2
Figure 1: The Z3 -maximal integral lattice-free polytopes with lattice width two. For
further reference, the polytopes are labeled by a pair of indices (i, j), where i is thenumber of facets and j the lattice diameter (defined at the end of the introduction).

The fivehollow
3-polytopesof withwidth
three
Figure 2: Figure:
The Z3 -maximal
integrallattice-freepolytopes
latticewidth three.
Proof strategy). 2017
In the proof of Theorem 1, weThe
useFinitenessa classificationof all Z2 -maximal polytopes inJanuaryP( 12 Zd18,
Threshold Width

Jan Hofmann (FU Berlin – BMS)

The case d = 4
Theoremw ∞ (4) = 2
Figure 1: The Z3 -maximal integral lattice-free polytopes with lattice width two. For
further reference, the polytopes are labeled by a pair of indices (i, j), where i is thenumber of facets and j the lattice diameter (defined at the end of the introduction).

The fivehollow
3-polytopesof withwidth
three
Figure 2: Figure:
The Z3 -maximal
integrallattice-freepolytopes
latticewidth three.
Proof strategy

To show: These polytopes have only f.m. lifts of bounded size.

). 2017
In the proof of Theorem 1, weThe
useFinitenessa classificationof all Z2 -maximal polytopes inJanuaryP( 12 Zd18,
Threshold Width

Jan Hofmann (FU Berlin – BMS)Polytopes with finitely many lifts of bounded size
Lemma
Let Q be a lattice pyramid with basis F and apex v . If F has finitely many
lifts of bounded size, then so does Q.
Proof:
It is enough to look at tight lifts, where a lift is called tight if there isa bijection between the vertices of P and Q.

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Polytopes with finitely many lifts of bounded size
Lemma
Let Q be a lattice pyramid with basis F and apex v . If F has finitely many
lifts of bounded size, then so does Q.
Proof:
It is enough to look at tight lifts, where a lift is called tight if there isa bijection between the vertices of P and Q.
Any tight lift of Q is of the form P(F̃ , h) := conv(F̃ ∪ {ṽ }), where F̃
is a tight lift of F and ṽ = (v , h) is a point in the fiber of v .

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Polytopes with finitely many lifts of bounded size
Lemma
Let Q be a lattice pyramid with basis F and apex v . If F has finitely many
lifts of bounded size, then so does Q.
Proof:
It is enough to look at tight lifts, where a lift is called tight if there isa bijection between the vertices of P and Q.
Any tight lift of Q is of the form P(F̃ , h) := conv(F̃ ∪ {ṽ }), where F̃
is a tight lift of F and ṽ = (v , h) is a point in the fiber of v .
Let m be the distance of v to F .

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Polytopes with finitely many lifts of bounded size
Lemma
Let Q be a lattice pyramid with basis F and apex v . If F has finitely many
lifts of bounded size, then so does Q.
Proof:
It is enough to look at tight lifts, where a lift is called tight if there isa bijection between the vertices of P and Q.
Any tight lift of Q is of the form P(F̃ , h) := conv(F̃ ∪ {ṽ }), where F̃
is a tight lift of F and ṽ = (v , h) is a point in the fiber of v .
Let m be the distance of v to F .
P(F̃ , h) is equivalent to P(F̃ , h + m) for all h ∈ Z.

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Polytopes with finitely many lifts of bounded size
Lemma
Let Q be a lattice pyramid with basis F and apex v . If F has finitely many
lifts of bounded size, then so does Q.
Proof:
It is enough to look at tight lifts, where a lift is called tight if there isa bijection between the vertices of P and Q.
Any tight lift of Q is of the form P(F̃ , h) := conv(F̃ ∪ {ṽ }), where F̃
is a tight lift of F and ṽ = (v , h) is a point in the fiber of v .
Let m be the distance of v to F .
P(F̃ , h) is equivalent to P(F̃ , h + m) for all h ∈ Z.

Hence there are at most m − 1 values of h that give non-equivalent
tight liftings P(F̃ , h), for any fixed F̃ .
Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Polytopes with finitely many lifts of bounded size

Lemma
Let Q be a lattice pyramid with basis F and apex v . If F has finitely many
lifts of bounded size, then so does Q.

Corollary
Lattice simplices have finitely many lifts of bounded size.

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Polytopes with finitely many lifts of bounded size

Lemma
Let Q be a lattice pyramid with basis F and apex v . If F has finitely many
lifts of bounded size, then so does Q.

Corollary
Lattice simplices have finitely many lifts of bounded size.
True by an inductive argument, as 1-simplices have only finitely many
lifts.

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

The case d = 4
Theoremw ∞ (4) = 2
Figure 1: The Z3 -maximal integral lattice-free polytopes with lattice width two. For
further reference, the polytopes are labeled by a pair of indices (i, j), where i is thenumber of facets and j the lattice diameter (defined at the end of the introduction).

The fivehollow
3-polytopesof withwidth
three
Figure 2: Figure:
The Z3 -maximal
integrallattice-freepolytopes
latticewidth three.
Proof strategy
Jan Hofmann
Berlin– BMS) 1, weTheThresholdWidth). 2017
In the(FUproofof Theorem
useFinitenessa classificationof allZ2 -maximal polytopes inJanuaryP( 1 Zd18,

Open problems

Theorem
12

w ∞ (d, n) ≤ w ∞ (d, n + 1) for all d, n.
w ∞ (d) ≤ w ∞ (d + 1) for all d.

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

Open problems

Theorem
12

w ∞ (d, n) ≤ w ∞ (d, n + 1) for all d, n.
w ∞ (d) ≤ w ∞ (d + 1) for all d.

Questions
Is w ∞ (d, n) ≤ w ∞ (d + 1, n + 1)? (True for n = d + 1 [HZ00])
Is w ∞ (d) = w ∞ (d, d + 1)? (True for d ≤ 4)

Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017

All of this is joint work with

Paco
MónicaChristian
The finiteness threshold width of lattice polytopes
arXiv:1607.00798
Jan Hofmann (FU Berlin – BMS)The Finiteness Threshold Width

January 18, 2017