Computational Approachesto Lattice Packing and Covering Problems

Frank Vallentin∗

01/03/2004

∗Joint work with Mathieu Dutour and Achill Schurmann

Sphere Coverings in Dimension 3

Z3

A3, the fcc-lattice A∗3, the bcc-lattice

Overview

I Space of triangulations

. finite point sets

. infinite periodic point sets

I Modern convex optimization: the MAXDET problem

I Application to lattice problems

. lattice covering problem

. (simultaneous) lattice packing-covering problem

I Results in low dimensions

I Open ends

Regular Triangulations & Secondary Polytopes (I)Gel’fand, Kapranov, Zelevinsky (1994)

Regular Triangulations & Secondary Polytopes (I)Gel’fand, Kapranov, Zelevinsky (1994)

I finite point set A ⊆ Rd

A

B C

DE

Regular Triangulations & Secondary Polytopes (I)Gel’fand, Kapranov, Zelevinsky (1994)

I finite point set A ⊆ Rd

A

B C

DE

I height function ω : A → R≥0ω(A) = 6 ω(B) = 3 ω(C) = 5ω(D) = 4 ω(E) = 2

Regular Triangulations & Secondary Polytopes (I)Gel’fand, Kapranov, Zelevinsky (1994)

I finite point set A ⊆ Rd

A

B C

DE

I height function ω : A → R≥0ω(A) = 6 ω(B) = 3 ω(C) = 5ω(D) = 4 ω(E) = 2

I lifting lω :

{A → Rd × R≥0

a 7→ (a, ω(a))

defines regular subdivision T (ω)

Regular Triangulations & Secondary Polytopes (I)Gel’fand, Kapranov, Zelevinsky (1994)

I finite point set A ⊆ Rd

A

B C

DE

I height function ω : A → R≥0ω(A) = 6 ω(B) = 3 ω(C) = 5ω(D) = 4 ω(E) = 2

I lifting lω :

{A → Rd × R≥0

a 7→ (a, ω(a))

defines regular subdivision T (ω)

I secondary cone ∆(T (ω)) = {ω′ ∈ RA≥0 : T (ω′) = T (ω)} is polyhedral cone

Regular Triangulations & Secondary Polytopes (I)Gel’fand, Kapranov, Zelevinsky (1994)

I finite point set A ⊆ Rd

A

B C

DE

I height function ω : A → R≥0ω(A) = 6 ω(B) = 3 ω(C) = 5ω(D) = 4 ω(E) = 2

I lifting lω :

{A → Rd × R≥0

a 7→ (a, ω(a))

defines regular subdivision T (ω)

I secondary cone ∆(T (ω)) = {ω′ ∈ RA≥0 : T (ω′) = T (ω)} is polyhedral cone

I dim ∆(T (ω)) = #A ⇐⇒ T (ω) is triangulation

Regular Triangulations & Secondary Polytopes (I)Gel’fand, Kapranov, Zelevinsky (1994)

I finite point set A ⊆ Rd

A

B C

DE

I height function ω : A → R≥0ω(A) = 6 ω(B) = 3 ω(C) = 5ω(D) = 4 ω(E) = 2

I lifting lω :

{A → Rd × R≥0

a 7→ (a, ω(a))

defines regular subdivision T (ω)

I secondary cone ∆(T (ω)) = {ω′ ∈ RA≥0 : T (ω′) = T (ω)} is polyhedral cone

I dim ∆(T (ω)) = #A ⇐⇒ T (ω) is triangulation

I facets of ∆(T (ω)) are defined by special interior (d− 1)-cells

Regular Triangulations & Secondary Polytopes (I)Gel’fand, Kapranov, Zelevinsky (1994)

I finite point set A ⊆ Rd

A

B C

DE

I height function ω : A → R≥0ω(A) = 6 ω(B) = 3 ω(C) = 5ω(D) = 4 ω(E) = 2

I lifting lω :

{A → Rd × R≥0

a 7→ (a, ω(a))

defines regular subdivision T (ω)

I secondary cone ∆(T (ω)) = {ω′ ∈ RA≥0 : T (ω′) = T (ω)} is polyhedral cone

I dim ∆(T (ω)) = #A ⇐⇒ T (ω) is triangulation

I facets of ∆(T (ω)) are defined by special interior (d− 1)-cells

I secondary fan: face-to-face tiling of RA≥0 by secondary cones

Regular Triangulations & Secondary Polytopes (II)

I Regular triangulations whose secondary cones share a facet differ by aflip.

Regular Triangulations & Secondary Polytopes (II)

I Regular triangulations whose secondary cones share a facet differ by aflip.

Regular Triangulations & Secondary Polytopes (II)

I Regular triangulations whose secondary cones share a facet differ by aflip.

Regular Triangulations & Secondary Polytopes (II)

I Regular triangulations whose secondary cones share a facet differ by aflip.

I Flips occur in polytopes with d + 2 vertices.

Regular Triangulations & Secondary Polytopes (II)

I Regular triangulations whose secondary cones share a facet differ by aflip.

I Flips occur in polytopes with d + 2 vertices.

I A polytope V with d + 2 vertices has exactly two different triangu-lations: conv(V \{v}), αv > 0, and conv(V \{v}), αv < 0, where∑v∈vert V

αvv = 0,∑

v∈vert V

αv = 0, is the affine relation of the vertices

Regular Triangulations & Secondary Polytopes (II)

I Regular triangulations whose secondary cones share a facet differ by aflip.

I Flips occur in polytopes with d + 2 vertices.

I A polytope V with d + 2 vertices has exactly two different triangu-lations: conv(V \{v}), αv > 0, and conv(V \{v}), αv < 0, where∑v∈vert V

αvv = 0,∑

v∈vert V

αv = 0, is the affine relation of the vertices

+

−

−

+

Regular Triangulations & Secondary Polytopes (III)

I One more example: Triangulating convex n-gons — The Associahedron

Regular Triangulations & Secondary Polytopes (IV)

Theorem. (Gel’fand, Kapranov, Zelevinsky)Let A ⊆ Rd be a finite point set. There is a (|A| − (d + 1))-dimensionalpolytope (the secondary polytope ΣA) whose normal fan equals the second-ary fan of A.

Regular Triangulations & Secondary Polytopes (IV)

Theorem. (Gel’fand, Kapranov, Zelevinsky)Let A ⊆ Rd be a finite point set. There is a (|A| − (d + 1))-dimensionalpolytope (the secondary polytope ΣA) whose normal fan equals the second-ary fan of A.

normal fan of a polytope

Regular Triangulations & Secondary Polytopes (IV)

Theorem. (Gel’fand, Kapranov, Zelevinsky)Let A ⊆ Rd be a finite point set. There is a (|A| − (d + 1))-dimensionalpolytope (the secondary polytope ΣA) whose normal fan equals the second-ary fan of A.

normal fan of a polytope

. vertices ⇔ regular triangulations

. edges ⇔ flips

. faces ⇔ regular subdivisions

Voronoi’s Reduction Theory (I)Voronoi (1908)

Voronoi’s Reduction Theory (I)Voronoi (1908)

I infinite point set Zd

Voronoi’s Reduction Theory (I)Voronoi (1908)

I infinite point set Zd

I positive semidefinite quadratic form Q ∈ Sd≥0

Q =

(2 −1−1 2

),

(0, 0)Q(0, 0)t = 0, (1, 0)Q(1, 0)t = 2, . . .

Voronoi’s Reduction Theory (I)Voronoi (1908)

I infinite point set Zd

I positive semidefinite quadratic form Q ∈ Sd≥0

Q =

(2 −1−1 2

),

(0, 0)Q(0, 0)t = 0, (1, 0)Q(1, 0)t = 2, . . .

I lifting lQ :

{Zd → Rd × R≥0

v 7→ (v, vtQv)

defines Delone subdivision Del(Q)

Voronoi’s Reduction Theory (I)Voronoi (1908)

I infinite point set Zd

I positive semidefinite quadratic form Q ∈ Sd≥0

Q =

(2 −1−1 2

),

(0, 0)Q(0, 0)t = 0, (1, 0)Q(1, 0)t = 2, . . .

I lifting lQ :

{Zd → Rd × R≥0

v 7→ (v, vtQv)

defines Delone subdivision Del(Q)

I secondary cone (L-type domain) ∆(Del(Q)) = {Q′ ∈ Sd≥0 : Del(Q′) = Del(Q)} is polyhedral cone

Voronoi’s Reduction Theory (I)Voronoi (1908)

I infinite point set Zd

I positive semidefinite quadratic form Q ∈ Sd≥0

Q =

(2 −1−1 2

),

(0, 0)Q(0, 0)t = 0, (1, 0)Q(1, 0)t = 2, . . .

I lifting lQ :

{Zd → Rd × R≥0

v 7→ (v, vtQv)

defines Delone subdivision Del(Q)

I secondary cone (L-type domain) ∆(Del(Q)) = {Q′ ∈ Sd≥0 : Del(Q′) = Del(Q)} is polyhedral cone

I dim ∆(Del(Q)) = dimSd≥0 = d(d+1)

2 ⇐⇒ Del(Q) is triangulation

Voronoi’s Reduction Theory (I)Voronoi (1908)

I infinite point set Zd

I positive semidefinite quadratic form Q ∈ Sd≥0

Q =

(2 −1−1 2

),

(0, 0)Q(0, 0)t = 0, (1, 0)Q(1, 0)t = 2, . . .

I lifting lQ :

{Zd → Rd × R≥0

v 7→ (v, vtQv)

defines Delone subdivision Del(Q)

I secondary cone (L-type domain) ∆(Del(Q)) = {Q′ ∈ Sd≥0 : Del(Q′) = Del(Q)} is polyhedral cone

I dim ∆(Del(Q)) = dimSd≥0 = d(d+1)

2 ⇐⇒ Del(Q) is triangulation

I facets of ∆(Del(Q)) are defined by special (d− 1)-cells

Voronoi’s Reduction Theory (I)Voronoi (1908)

I infinite point set Zd

I positive semidefinite quadratic form Q ∈ Sd≥0

Q =

(2 −1−1 2

),

(0, 0)Q(0, 0)t = 0, (1, 0)Q(1, 0)t = 2, . . .

I lifting lQ :

{Zd → Rd × R≥0

v 7→ (v, vtQv)

defines Delone subdivision Del(Q)

I secondary cone (L-type domain) ∆(Del(Q)) = {Q′ ∈ Sd≥0 : Del(Q′) = Del(Q)} is polyhedral cone

I dim ∆(Del(Q)) = dimSd≥0 = d(d+1)

2 ⇐⇒ Del(Q) is triangulation

I facets of ∆(Del(Q)) are defined by special (d− 1)-cells

I secondary fan: face-to-face tiling of Sd≥0 by secondary cones

Voronoi’s Reduction Theory (II)I Delone triangulations whose secondary cones share a facet differ by a

flip.

Voronoi’s Reduction Theory (II)I Delone triangulations whose secondary cones share a facet differ by a

flip.

Voronoi’s Reduction Theory (II)I Delone triangulations whose secondary cones share a facet differ by a

flip.

I The group GLd(Z) acts on the set of Delone triangulations as well ason the set of secondary cones.

Voronoi’s Reduction Theory (III)

I Theorem. (Voronoi)Under the group action, there are only finitely many non-equivalentDelone triangulations.

This implies that the union of all non-equivalent secondary cones givesa fundamental domain of Sd

≥0/GLd(Z).

Voronoi’s Reduction Theory (III)

I Theorem. (Voronoi)Under the group action, there are only finitely many non-equivalentDelone triangulations.

This implies that the union of all non-equivalent secondary cones givesa fundamental domain of Sd

≥0/GLd(Z).

Question. Does there also exist a “secondary polytope” in this setting?

MAXDET Optimization

minimize ctx− log det G(x)subject to G(x) = G0 + x1G1 + · · · + xdGd positive definite,

F (x) = F0 + x1F1 + · · · + xdFd positive semidefinite,Gi ∈ Sm, Fi ∈ Sn

MAXDET Optimization

minimize ctx− log det G(x)subject to G(x) = G0 + x1G1 + · · · + xdGd positive definite,

F (x) = F0 + x1F1 + · · · + xdFd positive semidefinite,Gi ∈ Sm, Fi ∈ Sn

I Minkowski (1905):MAXDET is a convex programming problem.

I Nesterov, Nemirovskii (1994):There exists an efficient algorithm (reduction to semidefinite program-ming), which solves MAXDET up to arbitrary precision

I Vandenberghe, Boyd, Wu (1998):Efficient implementation of a direct interior point method.

Application to Lattice Covering Problem (I)

I inhomogeneous minimum: µ(Q) = maxx∈Rd

minv∈Zd

Q[x− v]

.√

µ(Q) is maximum circumradius of the Delone polytopes in Del(Q).

. spheres with radius√

µ(Q) give the minimum covering of Rd

Application to Lattice Covering Problem (I)

I inhomogeneous minimum: µ(Q) = maxx∈Rd

minv∈Zd

Q[x− v]

.√

µ(Q) is maximum circumradius of the Delone polytopes in Del(Q).

. spheres with radius√

µ(Q) give the minimum covering of Rd

I covering density:

Θ(Q) =vol(sphere with radius

õ(Q))

vol(fundamental domain of Zd)=

õ(Q)d

det Q· vol Bd

Application to Lattice Covering Problem (I)

I inhomogeneous minimum: µ(Q) = maxx∈Rd

minv∈Zd

Q[x− v]

.√

µ(Q) is maximum circumradius of the Delone polytopes in Del(Q).

. spheres with radius√

µ(Q) give the minimum covering of Rd

I covering density:

Θ(Q) =vol(sphere with radius

õ(Q))

vol(fundamental domain of Zd)=

õ(Q)d

det Q· vol Bd

I Lattice Covering Problem:Determine Q ∈ Sd

>0 with Θ(Q) = minQ′∈Sd>0

Θ(Q′).

Application to Lattice Covering Problem (I)

I inhomogeneous minimum: µ(Q) = maxx∈Rd

minv∈Zd

Q[x− v]

.√

µ(Q) is maximum circumradius of the Delone polytopes in Del(Q).

. spheres with radius√

µ(Q) give the minimum covering of Rd

I covering density:

Θ(Q) =vol(sphere with radius

õ(Q))

vol(fundamental domain of Zd)=

õ(Q)d

det Q· vol Bd

I Lattice Covering Problem:Determine Q ∈ Sd

>0 with Θ(Q) = minQ′∈Sd>0

Θ(Q′).

. Strategy:Suppose µ(Q) = 1 and maximize det Q. Possible, because Θ(Q) =Θ(αQ). Even suppose that µ(Q) ≤ 1.

Application to Lattice Covering Problem (II)

I Lemma. Q ∈ Sd>0, L = conv{0, v1, . . . ,vd} d-dimensional simplex.

The circumradius (resp. Q) of L is at most 1⇐⇒

4 vt1Qv1 vt

2Qv2 . . . vtdQvd

vt1Qv1 vt

1Qv1 vt2Qv1 . . . vt

1Qvd... ... ... ...

vtdQvd vt

dQv1 vtdQv1 . . . vt

dQvd

is positive semidefinite

Application to Lattice Covering Problem (II)

I Lemma. Q ∈ Sd>0, L = conv{0, v1, . . . ,vd} d-dimensional simplex.

The circumradius (resp. Q) of L is at most 1⇐⇒

4 vt1Qv1 vt

2Qv2 . . . vtdQvd

vt1Qv1 vt

1Qv1 vt2Qv1 . . . vt

1Qvd... ... ... ...

vtdQvd vt

dQv1 vtdQv1 . . . vt

dQvd

is positive semidefinite

. This condition is linear in the entries of Q.

Application to Lattice Covering Problem (II)

I Lemma. Q ∈ Sd>0, L = conv{0, v1, . . . ,vd} d-dimensional simplex.

The circumradius (resp. Q) of L is at most 1⇐⇒

4 vt1Qv1 vt

2Qv2 . . . vtdQvd

vt1Qv1 vt

1Qv1 vt2Qv1 . . . vt

1Qvd... ... ... ...

vtdQvd vt

dQv1 vtdQv1 . . . vt

dQvd

is positive semidefinite

. This condition is linear in the entries of Q.

I Lattice Covering Problem restricted to full-dimensional secondary cones∆, is equivalent to a MAXDET problem: min

Q∈∆,µ(Q)≤1− log det(Q)

. Q ∈ ∆ are linear conditions(every facet of ∆ gives one condition).

. µ(Q) ≤ 1 are semidefinite conditions(every d-dimensional simplex of Del(Q) gives one condition).

Application to Lattice Packing-Covering Problem (I)

I homogeneous minimum: λ(Q) = minv∈Zd\{0}

Q[v]

. 12

√λ(Q) is maximum radius of spheres giving a sphere packing of Rd.

Application to Lattice Packing-Covering Problem (I)

I homogeneous minimum: λ(Q) = minv∈Zd\{0}

Q[v]

. 12

√λ(Q) is maximum radius of spheres giving a sphere packing of Rd.

I packing-covering constant: γ(Q) =√

µ(Q)12λ(Q)

Application to Lattice Packing-Covering Problem (I)

I homogeneous minimum: λ(Q) = minv∈Zd\{0}

Q[v]

. 12

√λ(Q) is maximum radius of spheres giving a sphere packing of Rd.

I packing-covering constant: γ(Q) =√

µ(Q)12λ(Q)

I (Simultaneous) Lattice Packing-Covering Problem:Determine Q ∈ Sd

>0 with γ(Q) = minQ′∈Sd>0

γ(Q′).

Application to Lattice Packing-Covering Problem (I)

I homogeneous minimum: λ(Q) = minv∈Zd\{0}

Q[v]

. 12

√λ(Q) is maximum radius of spheres giving a sphere packing of Rd.

I packing-covering constant: γ(Q) =√

µ(Q)12λ(Q)

I (Simultaneous) Lattice Packing-Covering Problem:Determine Q ∈ Sd

>0 with γ(Q) = minQ′∈Sd>0

γ(Q′).

Application to Lattice Packing-Covering Problem (I)

I homogeneous minimum: λ(Q) = minv∈Zd\{0}

Q[v]

. 12

√λ(Q) is maximum radius of spheres giving a sphere packing of Rd.

I packing-covering constant: γ(Q) =√

µ(Q)12λ(Q)

I (Simultaneous) Lattice Packing-Covering Problem:Determine Q ∈ Sd

>0 with γ(Q) = minQ′∈Sd>0

γ(Q′).

. Open Problem. Does there exist d with γ(Q) ≥ 2 for all Q ∈ Sd>0?

Application to Lattice Packing-Covering Problem (II)

I Strategy: maximize λ(Q) subject to µ(Q) ≤ 1.

γ(Q) =

√µ(Q)12λ(Q)

Application to Lattice Packing-Covering Problem (II)

I Strategy: maximize λ(Q) subject to µ(Q) ≤ 1.

γ(Q) =

√µ(Q)12λ(Q)

I Lattice Packing-Covering Problem restricted to subcones offull-dimensional secondary cones ∆, is a MAXDET problem:

minQ∈∆,µ(Q)≤1,λ(Q)=Q[v]

−λ(Q)

. Q ∈ ∆ are linear conditions(every facet of ∆ gives one condition)

. µ(Q) ≤ 1 are semidefinite conditions(every d-dimensional simplex of Del(Q) gives one condition).

. λ(Q) = Q[v] are linear conditions.(every edge of the Delone triangulation gives one condition)

Results in Low Dimensions (I)

I Implementation (C++) finds all locally optimal lattice coverings up todimension 5

dimension d 2 3 4 5# local optima 1 1 3 222

Results in Low Dimensions (I)

I Implementation (C++) finds all locally optimal lattice coverings up todimension 5

dimension d 2 3 4 5# local optima 1 1 3 222

I This verifies and extends known results

dimension d2 Kershner (1939)3 Bambah (1954)4 Delone, Ryshkov (1963)5 Ryshkov, Baranovskii (1975)

Results in Low Dimensions (I)

I Implementation (C++) finds all locally optimal lattice coverings up todimension 5

dimension d 2 3 4 5# local optima 1 1 3 222

I This verifies and extends known results

dimension d2 Kershner (1939)3 Bambah (1954)4 Delone, Ryshkov (1963)5 Ryshkov, Baranovskii (1975)

I In all cases the globally optimal lattice covering is given by Voronoi’sprincipal form (the lattice A∗

d)

xtQx = d

d∑i=1

x2i −

∑i6=j

xixj

Results in Low Dimensions (II)

I Ryshkov (1967):What is the first dimension d where Voronoi’s principal formdoes not give the best covering?

Results in Low Dimensions (II)

I Ryshkov (1967):What is the first dimension d where Voronoi’s principal formdoes not give the best covering?

I Ryshkov (1967): d ≤ 114

I Conway, Parker, Sloane (1980): d ≤ 24

I Smith (1988): d ≤ 22

I Baranovskii (1994): d ≤ 9

Results in Low Dimensions (III)

I Problem 1. There are probably several million non-equivalent locallyoptimal lattice coverings in dimension 6.

. Where are the good ones???

I Problem 2. Computation of one locally optimal lattice covering takesseveral minutes.

Results in Low Dimensions (III)

I Problem 1. There are probably several million non-equivalent locallyoptimal lattice coverings in dimension 6.

. Where are the good ones???

I Problem 2. Computation of one locally optimal lattice covering takesseveral minutes.

I Linear relaxation of the problem gives a local lower bound which we cancompute fast

I Use a combination of random walks and greedy optimization in flip graphof Delone triangulation.

Results in Low Dimensions (IV)

I Local Lower Bound

. L = conv{v1, . . . ,vd+1} d-dimensional simplex, c circumcenter of L,R circumradius of L, m centroid of L, then

R2 = dist(c, m)2 +1

d + 1

d+1∑i=1

dist(vi, m)2

Results in Low Dimensions (IV)

I Local Lower Bound

. L = conv{v1, . . . ,vd+1} d-dimensional simplex, c circumcenter of L,R circumradius of L, m centroid of L, then

R2 = dist(c, m)2 +1

d + 1

d+1∑i=1

dist(vi, m)2

.

R2 ≥ 1

d + 1

d+1∑i=1

dist(vi, m)2 = IL(Q)

Results in Low Dimensions (IV)

I Local Lower Bound

. L = conv{v1, . . . ,vd+1} d-dimensional simplex, c circumcenter of L,R circumradius of L, m centroid of L, then

R2 = dist(c, m)2 +1

d + 1

d+1∑i=1

dist(vi, m)2

.

R2 ≥ 1

d + 1

d+1∑i=1

dist(vi, m)2 = IL(Q)

. L1, . . . , Ln non-equivalent simplices of Delone triangulation D.

µ(Q) ≥ 1

n

n∑i=1

ILi(Q) = ID(Q)

Results in Low Dimensions (IV)

I Local Lower Bound

. L = conv{v1, . . . ,vd+1} d-dimensional simplex, c circumcenter of L,R circumradius of L, m centroid of L, then

R2 = dist(c, m)2 +1

d + 1

d+1∑i=1

dist(vi, m)2

.

R2 ≥ 1

d + 1

d+1∑i=1

dist(vi, m)2 = IL(Q)

. L1, . . . , Ln non-equivalent simplices of Delone triangulation D.

µ(Q) ≥ 1

n

n∑i=1

ILi(Q) = ID(Q)

. Maximization of ID(Q) over positive definite quadratic forms with de-terminant 1 gives lower bound for covering density

Results in Low Dimensions (V)0.50025

0.497740.49605

0.49497

0.49386

0.49247 0.49072

0.48902

0.48728

0.48589

0.48451

0.48244

0.480440.47855

0.47524

0.47305

0.47132

0.46800

0.46607

0.46212

0.45754

0.45710

0.45661

0.45616

0.45570

0.45524

0.45479

0.45434

0.45390

0.45346

0.453020.45258

0.45215

0.45171

0.45128

0.45085

0.45042

0.449910.44984

0.44999

0.44976

0.44966

0.44952

0.44947

0.44943

0.44935

0.44921

0.44904 0.44883

0.44856

Results in Low Dimensions (VI)

I The corresponding positive definite quadratic form is

Q26 ≈

1.9982 0.5270 −0.4170 −0.5270 0.5270 −1.05410.5270 1.9982 −0.4170 −0.5270 0.5270 −1.0541−0.4170 −0.4170 2.1082 −1.0541 −0.4170 0.8341−0.5270 −0.5270 −1.0541 1.9982 −0.5270 −0.41700.5270 0.5270 −0.4170 −0.5270 1.9982 −1.0541−1.0541 −1.0541 0.8341 −0.4170 −1.0541 2.1082

with covering density Θ(Q2

6) ≈ 0.477282 · vol B6

. Θ(Q) ≈ 0.493668 · vol B6

Results in Low Dimensions (VI)

I The corresponding positive definite quadratic form is

Q26 ≈

1.9982 0.5270 −0.4170 −0.5270 0.5270 −1.05410.5270 1.9982 −0.4170 −0.5270 0.5270 −1.0541−0.4170 −0.4170 2.1082 −1.0541 −0.4170 0.8341−0.5270 −0.5270 −1.0541 1.9982 −0.5270 −0.41700.5270 0.5270 −0.4170 −0.5270 1.9982 −1.0541−1.0541 −1.0541 0.8341 −0.4170 −1.0541 2.1082

with covering density Θ(Q2

6) ≈ 0.477282 · vol B6

. Θ(Q) ≈ 0.493668 · vol B6

I But after a while

Q16 ≈

2.0550 −0.9424 1.1126 0.2747 −0.9424 −0.6153−0.9424 1.9227 −0.5773 −0.7681 0.3651 −0.36511.1126 −0.5773 2.0930 −0.4934 −0.5773 −0.98040.2747 −0.7681 −0.4934 1.7550 −0.7681 0.7681−0.9424 0.3651 −0.5773 −0.7681 1.9227 −0.3651−0.6153 −0.3651 −0.9804 0.7681 −0.3651 1.9227

came up with Θ(Q1

6) ≈ 0.476962 · vol B6

Results in Low Dimensions (VII)

I Implementation (C++) finds all locally optimal lattice packing-coveringsup to dimension 5

dimension d 2 3 4 5# local optima 1 1 3 45

Results in Low Dimensions (VII)

I Implementation (C++) finds all locally optimal lattice packing-coveringsup to dimension 5

dimension d 2 3 4 5# local optima 1 1 3 45

I This verifies and extends known results

dimension d2 Ryshkov (1974)3 Boroczky (1986)4 Horvath (1982)5 Horvath (1986)

Results in Low Dimensions (VII)

I Implementation (C++) finds all locally optimal lattice packing-coveringsup to dimension 5

dimension d 2 3 4 5# local optima 1 1 3 45

I This verifies and extends known results

dimension d2 Ryshkov (1974)3 Boroczky (1986)4 Horvath (1982)5 Horvath (1986)

I Question. (Lagarias & Pleasants (2002))Is γ monotone increasing with dimension?

Results in Low Dimensions (VIII)

I The positive definite quadratic form

Q36 ≈

2.0089 0.5155 0.5155 −0.5155 0.9778 0.51550.5155 2.0089 0.5155 −0.5155 −0.5155 0.97780.5155 0.5155 2.0089 −0.5155 −0.9778 −0.5155−0.5155 −0.5155 −0.5155 2.0089 −0.9778 −0.51550.9778 −0.5155 −0.5155 −0.9778 2.0089 0.97780.5155 −0.9778 0.5155 −0.5155 0.9778 2.0089

gives the best known lattice packing-covering with constant

γ(Q36) ≈ 1.411081 < γ(H5) ≈ 1.449456

Results in Low Dimensions (VIII)

I The positive definite quadratic form

Q36 ≈

2.0089 0.5155 0.5155 −0.5155 0.9778 0.51550.5155 2.0089 0.5155 −0.5155 −0.5155 0.97780.5155 0.5155 2.0089 −0.5155 −0.9778 −0.5155−0.5155 −0.5155 −0.5155 2.0089 −0.9778 −0.51550.9778 −0.5155 −0.5155 −0.9778 2.0089 0.97780.5155 −0.9778 0.5155 −0.5155 0.9778 2.0089

gives the best known lattice packing-covering with constant

γ(Q36) ≈ 1.411081 < γ(H5) ≈ 1.449456

I Q36 lies in the same secondary cone as Q2

6.

Results in Low Dimensions (IX)

I What is the message?

. What is the origin of these new sphere coverings?

. How to find exact solutions?

Results in Low Dimensions (IX)

I What is the message?

. What is the origin of these new sphere coverings?

. How to find exact solutions?

I Main Tool. Automorphism group of Delone triangulation is subgroupof local optima.

Results in Low Dimensions (IX)

I What is the message?

. What is the origin of these new sphere coverings?

. How to find exact solutions?

I Main Tool. Automorphism group of Delone triangulation is subgroupof local optima.

. |Aut(Del(Q26))| = |Aut(Del(Q3

6))| = 3840, |Aut(Del(Q16))| = 240.

Results in Low Dimensions (IX)

I What is the message?

. What is the origin of these new sphere coverings?

. How to find exact solutions?

I Main Tool. Automorphism group of Delone triangulation is subgroupof local optima.

. |Aut(Del(Q26))| = |Aut(Del(Q3

6))| = 3840, |Aut(Del(Q16))| = 240.

. Cone C1 in which Q26 and Q3

6 lie has has 130 facets and 7, 145, 429extreme rays.

. Cone C2 in which Q16 lies has 100 facets and 2, 257, 616 extreme rays.

Results in Low Dimensions (IX)

I What is the message?

. What is the origin of these new sphere coverings?

. How to find exact solutions?

I Main Tool. Automorphism group of Delone triangulation is subgroupof local optima.

. |Aut(Del(Q26))| = |Aut(Del(Q3

6))| = 3840, |Aut(Del(Q16))| = 240.

. Cone C1 in which Q26 and Q3

6 lie has has 130 facets and 7, 145, 429extreme rays.

. Cone C2 in which Q16 lies has 100 facets and 2, 257, 616 extreme rays.

. Both cones have one extreme ray associated to lattice E∗6.

Aut(Del(Q16)) ⊆ Aut(Del(Q2

6)) ⊆ Aut(Del(QE∗6))

Results in Low Dimensions (IX)

I What is the message?

. What is the origin of these new sphere coverings?

. How to find exact solutions?

I Main Tool. Automorphism group of Delone triangulation is subgroupof local optima.

. |Aut(Del(Q26))| = |Aut(Del(Q3

6))| = 3840, |Aut(Del(Q16))| = 240.

. Cone C1 in which Q26 and Q3

6 lie has has 130 facets and 7, 145, 429extreme rays.

. Cone C2 in which Q16 lies has 100 facets and 2, 257, 616 extreme rays.

. Both cones have one extreme ray associated to lattice E∗6.

Aut(Del(Q16)) ⊆ Aut(Del(Q2

6)) ⊆ Aut(Del(QE∗6))

. Invariant subspace I1 of Aut(Del(Q26)) has dimension 2.

. Invariant subspace I2 of Aut(Del(Q16)) has dimension 4.

Results in Low Dimensions (X)

I Unified View

QE∗6

R1

R2

Q16

Q26Q3

6

C1

C2

invariant subspace I1

boundary of S6≥0

I Theorem. Q16 gives the least dense lattice covering among all positive

definite quadratic forms whose Delone subdivision is a refinement ofDel(QE∗

6).

Results in Low Dimensions (XI)I Beautification. Finding exact coordinates boils down/up to solving a

system of polynomial equations.

Results in Low Dimensions (XI)I Beautification. Finding exact coordinates boils down/up to solving a

system of polynomial equations.

Q26 = QE∗

6+

√1057− 1

88R1

Q36 = QE∗

6+

√798− 18

79R1

Results in Low Dimensions (XI)I Beautification. Finding exact coordinates boils down/up to solving a

system of polynomial equations.

Q26 = QE∗

6+

√1057− 1

88R1

Q36 = QE∗

6+

√798− 18

79R1

Q16 = QE∗

6+ xR2 + yR3 + zR4

where

−35yz3 − 6y2z + 4y2z2 + 5y3z − 15z2 + 6y3 − 11yz − 25yz2 + 8y2 − 10z4 − 21z3 − 4z + 2y

14y3 − 31y2z − 6z3 + 6y − 11z + 20y2 − 34yz − 15z2 − 31yz2= x

3y4 + (−7z + 7)y + (−14z − 9z2 + 5)y2 + (−9− 9z3 − 14z2 + 1)y − 5z2 − 2z − 5z3 − 2z4 = 0

and z is the root near 0.138317657 of ∂Θ(z)∂z

where

Θ(z) =1

4096(72 + 1862xy2z + 2508xy2 + 264x + 520y + 861x2yz + 809xyz2 + 2208xyz + 941yz2

+1436xy3 + 788y2z2 + 1238yz + 223yz3 + 2143y2z + 1217y3z + 117xz3 + 487xz2 + 216x3y

+108x3z + 966x2y2 + 521x2z + 1166x2y + 193x2z2 + 634xz + 1424xy + 232z + 1380y2

+1622y3 + 270z2 + 22z4 + 132z3 + 332x2 + 710y4 + 144x3)6/

((6xy + 3xz + 4x + 12y + z2 + 4z + 7yz + 10y2 + 3)7(4x + 5y + 3z + 3)10)

Open Ends

I Challenge. Show that the best-known lattice covering is indeed op-timal.

Open Ends

I Challenge. Show that the best-known lattice covering is indeed op-timal.

I Challenge. Find a non-lattice sphere covering which is less dense thanevery lattice covering of the same dimension.

Open Ends

I Challenge. Show that the best-known lattice covering is indeed op-timal.

I Challenge. Find a non-lattice sphere covering which is less dense thanevery lattice covering of the same dimension.

I Use group theory and find an equivariant set-up (work in progress withMathieu Dutour)

Open Ends

I Challenge. Show that the best-known lattice covering is indeed op-timal.

I Challenge. Find a non-lattice sphere covering which is less dense thanevery lattice covering of the same dimension.

I Use group theory and find an equivariant set-up (work in progress withMathieu Dutour)

I Find good sphere coverings in dimension 7 and 8.

Open Ends

I Challenge. Show that the best-known lattice covering is indeed op-timal.

I Challenge. Find a non-lattice sphere covering which is less dense thanevery lattice covering of the same dimension.

I Use group theory and find an equivariant set-up (work in progress withMathieu Dutour)

I Find good sphere coverings in dimension 7 and 8.

I Use structural insights to show that the problem of determining thecovering radius is (at least) NP-hard.