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Colorful simplicial depth Zuzana Pat akov a joint work with Karim - - PowerPoint PPT Presentation

Nov 25, 2023 279 likes •1.62k views

Colorful simplicial depth Zuzana Pat akov a joint work with Karim Adiprasito, Philip Brinkmann, Arnau Padrol, Pavel Pat ak, and Raman Sanyal ICERM 1st December, Providence Colorful configurations C = { C 0 , . . . , C d } :

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SLIDE 1

Colorful simplicial depth

Zuzana Pat´ akov´ a joint work with Karim Adiprasito, Philip Brinkmann, Arnau Padrol, Pavel Pat´ ak, and Raman Sanyal ICERM 1st December, Providence

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SLIDE 2

Colorful configurations

  • C = {C0, . . . , Cd}
  • ϕ: Ci → Rd

⇒ (C, ϕ) is a colorful configuration

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SLIDE 3

Colorful configurations

  • C = {C0, . . . , Cd}
  • ϕ: Ci → Rd

⇒ (C, ϕ) is a colorful configuration Colorful simplex

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SLIDE 4

Colorful configurations

  • C = {C0, . . . , Cd}
  • ϕ: Ci → Rd

⇒ (C, ϕ) is a colorful configuration

  • T ⊆ Ci is colorful if it contains |T| colors

Colorful simplex

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SLIDE 5

Colorful configurations

  • C = {C0, . . . , Cd}
  • ϕ: Ci → Rd

⇒ (C, ϕ) is a colorful configuration

  • T ⊆ Ci is colorful if it contains |T| colors
  • a simplex is colorful if it is spanned by ϕ(T), T colorful

Colorful simplex

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SLIDE 6

Colorful configurations

  • C = {C0, . . . , Cd}
  • ϕ: Ci → Rd

⇒ (C, ϕ) is a colorful configuration

  • T ⊆ Ci is colorful if it contains |T| colors
  • a simplex is colorful if it is spanned by ϕ(T), T colorful

Colorful simplex

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SLIDE 7

Colorful configurations

  • C = {C0, . . . , Cd}
  • ϕ: Ci → Rd

⇒ (C, ϕ) is a colorful configuration

  • T ⊆ Ci is colorful if it contains |T| colors
  • a simplex is colorful if it is spanned by ϕ(T), T colorful

Colorful simplex

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SLIDE 8

Colorful configurations

  • C = {C0, . . . , Cd}
  • ϕ: Ci → Rd

⇒ (C, ϕ) is a colorful configuration

  • T ⊆ Ci is colorful if it contains |T| colors
  • a simplex is colorful if it is spanned by ϕ(T), T colorful

Colorful simplex

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SLIDE 9

Colorful configurations

  • C = {C0, . . . , Cd}
  • ϕ: Ci → Rd

⇒ (C, ϕ) is a colorful configuration

  • T ⊆ Ci is colorful if it contains |T| colors
  • a simplex is colorful if it is spanned by ϕ(T), T colorful

Not Colorful simplex

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SLIDE 10

Colorful configurations

  • C = {C0, . . . , Cd}
  • ϕ: Ci → Rd

⇒ (C, ϕ) is a colorful configuration

  • T ⊆ Ci is colorful if it contains |T| colors
  • a simplex is colorful if it is spanned by ϕ(T), T colorful

Not Colorful simplex

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SLIDE 11

Centered colorful configurations

  • p ∈ Rd arbitrary point . . . wlog p = 0
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SLIDE 12

Centered colorful configurations

  • p ∈ Rd arbitrary point . . . wlog p = 0
  • assume: 0 does not lie on a boundary of any colorful simplex
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SLIDE 13

Centered colorful configurations

  • p ∈ Rd arbitrary point . . . wlog p = 0
  • assume: 0 does not lie on a boundary of any colorful simplex

Allowed

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SLIDE 14

Centered colorful configurations

  • p ∈ Rd arbitrary point . . . wlog p = 0
  • assume: 0 does not lie on a boundary of any colorful simplex

Allowed

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SLIDE 15

Centered colorful configurations

  • p ∈ Rd arbitrary point . . . wlog p = 0
  • assume: 0 does not lie on a boundary of any colorful simplex

Not Allowed

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SLIDE 16

Centered colorful configurations

  • p ∈ Rd arbitrary point . . . wlog p = 0
  • assume: 0 does not lie on a boundary of any colorful simplex
  • C is centered . . . 0 ∈ conv ϕ(Ci)
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SLIDE 17

Centered colorful configurations

  • p ∈ Rd arbitrary point . . . wlog p = 0
  • assume: 0 does not lie on a boundary of any colorful simplex
  • C is centered . . . 0 ∈ conv ϕ(Ci)

Centered

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SLIDE 18

Centered colorful configurations

  • p ∈ Rd arbitrary point . . . wlog p = 0
  • assume: 0 does not lie on a boundary of any colorful simplex
  • C is centered . . . 0 ∈ conv ϕ(Ci)

Centered

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SLIDE 19

Centered colorful configurations

  • p ∈ Rd arbitrary point . . . wlog p = 0
  • assume: 0 does not lie on a boundary of any colorful simplex
  • C is centered . . . 0 ∈ conv ϕ(Ci)

Centered

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SLIDE 20

Centered colorful configurations

  • p ∈ Rd arbitrary point . . . wlog p = 0
  • assume: 0 does not lie on a boundary of any colorful simplex
  • C is centered . . . 0 ∈ conv ϕ(Ci)

Centered

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SLIDE 21

Centered colorful configurations

  • p ∈ Rd arbitrary point . . . wlog p = 0
  • assume: 0 does not lie on a boundary of any colorful simplex
  • C is centered . . . 0 ∈ conv ϕ(Ci)

Centered

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SLIDE 22

Centered colorful configurations

  • p ∈ Rd arbitrary point . . . wlog p = 0
  • assume: 0 does not lie on a boundary of any colorful simplex
  • C is centered . . . 0 ∈ conv ϕ(Ci)

Not Centered

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SLIDE 23

Centered colorful configurations

  • p ∈ Rd arbitrary point . . . wlog p = 0
  • assume: 0 does not lie on a boundary of any colorful simplex
  • C is centered . . . 0 ∈ conv ϕ(Ci)
  • colorful d-dim simplex S is hitting if 0 ∈ conv S

A colorful configuration

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SLIDE 24

Centered colorful configurations

  • p ∈ Rd arbitrary point . . . wlog p = 0
  • assume: 0 does not lie on a boundary of any colorful simplex
  • C is centered . . . 0 ∈ conv ϕ(Ci)
  • colorful d-dim simplex S is hitting if 0 ∈ conv S

Hitting simplex

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SLIDE 25

Centered colorful configurations

  • p ∈ Rd arbitrary point . . . wlog p = 0
  • assume: 0 does not lie on a boundary of any colorful simplex
  • C is centered . . . 0 ∈ conv ϕ(Ci)
  • colorful d-dim simplex S is hitting if 0 ∈ conv S

Another hitting simplex

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SLIDE 26

Colorful simplicial depth

  • C is a colorful centered configuration
  • colorful simplicial depth of C = cdepth(C)

= number of hitting simplices of C Deza, Huang, Stephen and Terlaky ’06

  • Placing all Ci the same simplicial depth . . . . . . . . . Liu ’90
  • points with maximal simplicial depth

≈ higher dim analogue of median

  • applications in statistics
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SLIDE 27

Colorful simplicial depth

  • C is a colorful centered configuration
  • colorful simplicial depth of C = cdepth(C)

= number of hitting simplices of C Deza, Huang, Stephen and Terlaky ’06

  • Placing all Ci the same simplicial depth . . . . . . . . . Liu ’90
  • points with maximal simplicial depth

≈ higher dim analogue of median

  • applications in statistics
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SLIDE 28

Colorful simplicial depth

  • C is a colorful centered configuration
  • colorful simplicial depth of C = cdepth(C)

= number of hitting simplices of C Deza, Huang, Stephen and Terlaky ’06

  • Placing all Ci the same simplicial depth . . . . . . . . . Liu ’90
  • points with maximal simplicial depth

≈ higher dim analogue of median

  • applications in statistics
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SLIDE 29

Known results

Theorem (Colorful Carath´ eodory, B´ ar´ any ’82)

There is always at least one hitting simplex. (cdepth ≥ 1)

Conjecture (Deza, Huang, Stephen, Terlaky, ’06)

If Card C0 = Card C1 = . . . = Card Cd = d + 1, then

1 cdepth C ≥ d2 + 1 2 cdepth C ≤ 1 + dd+1

Deza et al: both bounds can be attained Lower bound: Deza et al [’06], B´ ar´ any, Matouˇ sek [’07], . . . Sarrabezolles [’15]

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SLIDE 30

Known results

Theorem (Colorful Carath´ eodory, B´ ar´ any ’82)

There is always at least one hitting simplex. (cdepth ≥ 1)

Conjecture (Deza, Huang, Stephen, Terlaky, ’06)

If Card C0 = Card C1 = . . . = Card Cd = d + 1, then

1 cdepth C ≥ d2 + 1 2 cdepth C ≤ 1 + dd+1

Deza et al: both bounds can be attained Lower bound: Deza et al [’06], B´ ar´ any, Matouˇ sek [’07], . . . Sarrabezolles [’15]

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SLIDE 31

Known results

Theorem (Colorful Carath´ eodory, B´ ar´ any ’82)

There is always at least one hitting simplex. (cdepth ≥ 1)

Conjecture (Deza, Huang, Stephen, Terlaky, ’06)

If Card C0 = Card C1 = . . . = Card Cd = d + 1, then

1 cdepth C ≥ d2 + 1 2 cdepth C ≤ 1 + dd+1

Deza et al: both bounds can be attained Lower bound: Deza et al [’06], B´ ar´ any, Matouˇ sek [’07], . . . Sarrabezolles [’15]

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SLIDE 32

Known results

Theorem (Colorful Carath´ eodory, B´ ar´ any ’82)

There is always at least one hitting simplex. (cdepth ≥ 1)

Conjecture (Deza, Huang, Stephen, Terlaky, ’06)

If Card C0 = Card C1 = . . . = Card Cd = d + 1, then

1 cdepth C ≥ d2 + 1 2 cdepth C ≤ 1 + dd+1

Deza et al: both bounds can be attained Lower bound: Deza et al [’06], B´ ar´ any, Matouˇ sek [’07], . . . Sarrabezolles [’15]

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SLIDE 33

Known results

Theorem (Colorful Carath´ eodory, B´ ar´ any ’82)

There is always at least one hitting simplex. (cdepth ≥ 1)

Conjecture (Deza, Huang, Stephen, Terlaky, ’06)

If Card C0 = Card C1 = . . . = Card Cd = d + 1, then

1 cdepth C ≥ d2 + 1 2 cdepth C ≤ 1 + dd+1

Deza et al: both bounds can be attained Lower bound: Deza et al [’06], B´ ar´ any, Matouˇ sek [’07], . . . Sarrabezolles [’15]

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SLIDE 34

Upper bound

Theorem (Adiprasito, Brinkmann, Padrol, Pat´ ak, P, Sanyal)

cdepth C ≤ 1 +

d

  • i=0
  • Card Ci − 1
  • .
  • for Card Ci = d + 1, we have Deza’s upper bound 1 + dd+1
  • the bound is tight!
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SLIDE 35

Upper bound

Theorem (Adiprasito, Brinkmann, Padrol, Pat´ ak, P, Sanyal)

cdepth C ≤ 1 +

d

  • i=0
  • Card Ci − 1
  • .
  • for Card Ci = d + 1, we have Deza’s upper bound 1 + dd+1
  • the bound is tight!
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SLIDE 36

Upper bound

Theorem (Adiprasito, Brinkmann, Padrol, Pat´ ak, P, Sanyal)

cdepth C ≤ 1 +

d

  • i=0
  • Card Ci − 1
  • .
  • for Card Ci = d + 1, we have Deza’s upper bound 1 + dd+1
  • the bound is tight!
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SLIDE 37

Upper bound

Theorem (Adiprasito, Brinkmann, Padrol, Pat´ ak, P, Sanyal)

cdepth C ≤ 1 +

d

  • i=0
  • Card Ci − 1
  • .
  • for Card Ci = d + 1, we have Deza’s upper bound 1 + dd+1
  • the bound is tight!
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SLIDE 38

Upper bound

Theorem (Adiprasito, Brinkmann, Padrol, Pat´ ak, P, Sanyal)

cdepth C ≤ 1 +

d

  • i=0
  • Card Ci − 1
  • .
  • for Card Ci = d + 1, we have Deza’s upper bound 1 + dd+1
  • the bound is tight!
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SLIDE 39

Upper bound

Theorem (Adiprasito, Brinkmann, Padrol, Pat´ ak, P, Sanyal)

cdepth C ≤ 1 +

d

  • i=0
  • Card Ci − 1
  • .
  • for Card Ci = d + 1, we have Deza’s upper bound 1 + dd+1
  • the bound is tight!
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SLIDE 40

Topological reformulation

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SLIDE 41

Topological approach

  • A = abstract simpl. complex of all colorful sets in Ci
  • B = all sets S ⊂ Ci s.t. ϕ(S) is non-hitting
  • fi(K) = number of i-dim simplices in K
  • cdepth(C) = fd(A) − fd(B)
  • A(d−1) = B(d−1)

⇒ for i < d fi(A) = fi(B) ⇒ for i < d − 1

  • βi(A) =

βi(B)

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SLIDE 42

Topological approach

  • A = abstract simpl. complex of all colorful sets in Ci
  • B = all sets S ⊂ Ci s.t. ϕ(S) is non-hitting
  • fi(K) = number of i-dim simplices in K
  • cdepth(C) = fd(A) − fd(B)
  • A(d−1) = B(d−1)

⇒ for i < d fi(A) = fi(B) ⇒ for i < d − 1

  • βi(A) =

βi(B)

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SLIDE 43

Topological approach

  • A = abstract simpl. complex of all colorful sets in Ci
  • B = all sets S ⊂ Ci s.t. ϕ(S) is non-hitting
  • fi(K) = number of i-dim simplices in K
  • cdepth(C) = fd(A) − fd(B)
  • A(d−1) = B(d−1)

⇒ for i < d fi(A) = fi(B) ⇒ for i < d − 1

  • βi(A) =

βi(B)

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SLIDE 44

Topological approach

  • A = abstract simpl. complex of all colorful sets in Ci
  • B = all sets S ⊂ Ci s.t. ϕ(S) is non-hitting
  • fi(K) = number of i-dim simplices in K
  • cdepth(C) = fd(A) − fd(B)
  • A(d−1) = B(d−1)

⇒ for i < d fi(A) = fi(B) ⇒ for i < d − 1

  • βi(A) =

βi(B)

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SLIDE 45

Topological approach

  • A = abstract simpl. complex of all colorful sets in Ci
  • B = all sets S ⊂ Ci s.t. ϕ(S) is non-hitting
  • fi(K) = number of i-dim simplices in K
  • cdepth(C) = fd(A) − fd(B)
  • A(d−1) = B(d−1)

⇒ for i < d fi(A) = fi(B) ⇒ for i < d − 1

  • βi(A) =

βi(B)

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SLIDE 46

Topological approach

  • A = abstract simpl. complex of all colorful sets in Ci
  • B = all sets S ⊂ Ci s.t. ϕ(S) is non-hitting
  • fi(K) = number of i-dim simplices in K
  • cdepth(C) = fd(A) − fd(B)
  • A(d−1) = B(d−1)

⇒ for i < d fi(A) = fi(B) ⇒ for i < d − 1

  • βi(A) =

βi(B)

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SLIDE 47

Topological approach

  • A = abstract simpl. complex of all colorful sets in Ci
  • B = all sets S ⊂ Ci s.t. ϕ(S) is non-hitting
  • fi(K) = number of i-dim simplices in K
  • cdepth(C) = fd(A) − fd(B)
  • A(d−1) = B(d−1)

⇒ for i < d fi(A) = fi(B) ⇒ for i < d − 1

  • βi(A) =

βi(B)

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SLIDE 48

Topological approach

cdepth C = fd(A) − fd(B)

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SLIDE 49

Topological approach

cdepth C = fd(A) − fd(B)

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SLIDE 50

Topological approach

cdepth C = fd(A) − fd(B)

  • χ(A) = −1 +

d

  • i=0

(−1)ifi(A) =

d

  • i=0

(−1)i βi(A)

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SLIDE 51

Topological approach

cdepth C = fd(A) − fd(B)

  • χ(A) = −1 +

d

  • i=0

(−1)ifi(A) =

d

  • i=0

(−1)i βi(A) ⇒ fd(A) = (−1)d d

  • i=0

(−1)i βi(A) + 1 −

d−1

  • i=0

(−1)ifi(A)

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SLIDE 52

Topological approach

cdepth C = (−1)d d

  • i=0

(−1)i βi(A) + 1 −

d−1

  • i=0

(−1)ifi(A)

  • − fd(B)

⇒ fd(A) = (−1)d d

  • i=0

(−1)i βi(A) + 1 −

d−1

  • i=0

(−1)ifi(A)

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SLIDE 53

Topological approach

cdepth C = (−1)d d

  • i=0

(−1)i βi(A) + 1 −

d−1

  • i=0

(−1)ifi(A)

  • − fd(B)
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SLIDE 54

Topological approach

cdepth C = (−1)d d

  • i=0

(−1)i βi(A) + 1 −

d−1

  • i=0

(−1)ifi(A)

  • − fd(B)

fd(B) = (−1)d d

  • i=0

(−1)i βi(B) + 1 −

d−1

  • i=0

(−1)ifi(B)

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SLIDE 55

Topological approach

cdepth C = (−1)d d

  • i=0

(−1)i βi(A) + 1 −

d−1

  • i=0

(−1)ifi(A) −

d

  • i=0

(−1)i βi(B) − 1 +

d−1

  • i=0

(−1)ifi(B)

  • fd(B) = (−1)d

d

  • i=0

(−1)i βi(B) + 1 −

d−1

  • i=0

(−1)ifi(B)

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SLIDE 56

Topological approach

cdepth C = (−1)d d

  • i=0

(−1)i βi(A) + 1 −

d−1

  • i=0

(−1)ifi(A) −

d

  • i=0

(−1)i βi(B) − 1 +

d−1

  • i=0

(−1)ifi(B)

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SLIDE 57

Topological approach

cdepth C = (−1)d d

  • i=0

(−1)i βi(A)−

d−1

  • i=0

(−1)ifi(A) −

d

  • i=0

(−1)i βi(B)+

d−1

  • i=0

(−1)ifi(B)

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SLIDE 58

Topological approach

cdepth C = (−1)d d

  • i=0

(−1)i βi(A)−

d−1

  • i=0

(−1)ifi(A) −

d

  • i=0

(−1)i βi(B)+

d−1

  • i=0

(−1)ifi(B)

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SLIDE 59

Topological approach

cdepth C = (−1)d d

  • i=0

(−1)i βi(A)−

d−1

  • i=0

(−1)ifi(A) −

d

  • i=0

(−1)i βi(B)+

d−1

  • i=0

(−1)ifi(B)

  • For i < d: fi(A) = fi(B)
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SLIDE 60

Topological approach

cdepth C = (−1)d d

  • i=0

(−1)i βi(A) −

d

  • i=0

(−1)i βi(B)

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SLIDE 61

Topological approach

cdepth C = (−1)d d

  • i=0

(−1)i βi(A) −

d

  • i=0

(−1)i βi(B)

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SLIDE 62

Topological approach

cdepth C = (−1)d d

  • i=0

(−1)i βi(A) −

d

  • i=0

(−1)i βi(B)

  • For i < d − 1:

βi(A) = βi(B)

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SLIDE 63

Topological approach

cdepth C = (−1)d (−1)d βd(A) + (−1)d−1 βd−1(A) −(−1)d βd(B) − (−1)d−1 βd−1(B)

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SLIDE 64

Topological approach

cdepth C = (−1)d (−1)d βd(A) + (−1)d−1 βd−1(A) −(−1)d βd(B) − (−1)d−1 βd−1(B)

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SLIDE 65

Topological approach

cdepth C = βd(A)− βd−1(A) − βd(B) + βd−1(B)

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SLIDE 66

Topological approach

cdepth C = βd(A)− βd−1(A) − βd(B) + βd−1(B)

slide-67
SLIDE 67

Topological approach

cdepth C = βd(A) − 0 − βd(B) + βd−1(B)

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SLIDE 68

Topological approach

cdepth C = βd(A) − βd(B) + βd−1(B)

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SLIDE 69

Topological approach

cdepth C = βd(A) − βd(B) + βd−1(B)

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SLIDE 70

Topological approach

cdepth C =

d

  • i=0
  • |Ci| − 1
  • −

βd(B) + βd−1(B)

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SLIDE 71

Topological approach

cdepth C =

d

  • i=0
  • |Ci| − 1
  • −

βd(B) + βd−1(B)

slide-72
SLIDE 72

Topological approach

cdepth C =

d

  • i=0
  • |Ci| − 1
  • −

βd(B) + βd−1(B) Our main Lemma: βd−1(B) = 1

slide-73
SLIDE 73

Topological approach

cdepth C =

d

  • i=0
  • |Ci| − 1
  • −

βd(B) + 1

slide-74
SLIDE 74

Topological approach

cdepth C =

d

  • i=0
  • |Ci| − 1
  • −

βd(B) + 1 ⇒ cdepth C ≤

d

  • i=0
  • |Ci| − 1
  • + 1
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SLIDE 75

Main lemma

Lemma

  • βd−1(B; Z2) = 1.

Proof idea:

1 First show for a special configuration of points: 2 Use flips preserving

βd−1(B; Z2)

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SLIDE 76

Main lemma

Lemma

  • βd−1(B; Z2) = 1.

Proof idea:

1 First show for a special configuration of points: 2 Use flips preserving

βd−1(B; Z2)

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SLIDE 77

Main lemma

Lemma

  • βd−1(B; Z2) = 1.

Proof idea:

1 First show for a special configuration of points: 2 Use flips preserving

βd−1(B; Z2)

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SLIDE 78

Main lemma

Lemma

  • βd−1(B; Z2) = 1.

Proof idea:

1 First show for a special configuration of points: 2 Use flips preserving

βd−1(B; Z2)

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SLIDE 79

Further connections

slide-80
SLIDE 80

Further connections – normal surface theory

  • normal d-fan = collection of polyhedral cones
slide-81
SLIDE 81

Further connections – normal surface theory

  • normal d-fan = collection of polyhedral cones

1-fan, given by normals of a triangle

slide-82
SLIDE 82

Further connections – normal surface theory

  • normal d-fan = collection of polyhedral cones

1-fan, given by normals of a triangle

slide-83
SLIDE 83

Further connections – normal surface theory

  • normal d-fan = collection of polyhedral cones

2-fan; halfplanes = leafs

slide-84
SLIDE 84

Further connections – normal surface theory

  • normal d-fan = collection of polyhedral cones

2-fan; halfplanes = leafs

  • two (and more) normal d-fans ⇒ common refinement
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SLIDE 85

Further connections – normal surface theory

  • Setting: F1, . . . Fd−1 normal (d − 1)-fans in general position

with leafs LFi

1 , LFi 2 , LFi 3

common refinement = collection of rays LF1

i1 ∩ . . . ∩ LFd−1 id−1

  • Question: Max number of rays in the common refinement?
  • Conjecture (Burton’03): 1 + 2d−1
slide-86
SLIDE 86

Further connections – normal surface theory

  • Setting: F1, . . . Fd−1 normal (d − 1)-fans in general position

with leafs LFi

1 , LFi 2 , LFi 3

common refinement = collection of rays LF1

i1 ∩ . . . ∩ LFd−1 id−1

  • Question: Max number of rays in the common refinement?
  • Conjecture (Burton’03): 1 + 2d−1
slide-87
SLIDE 87

Further connections – normal surface theory

  • Setting: F1, . . . Fd−1 normal (d − 1)-fans in general position

with leafs LFi

1 , LFi 2 , LFi 3

common refinement = collection of rays LF1

i1 ∩ . . . ∩ LFd−1 id−1

  • Question: Max number of rays in the common refinement?
  • Conjecture (Burton’03): 1 + 2d−1
slide-88
SLIDE 88

Further connections – normal surface theory

  • P1, . . . , Pk ⊂ Rd be polytopes (not necessarily full dim)
  • Minkowski sum

P1 + P2 + . . . + Pk = {p1 + p2 + . . . + pk | pi ∈ Pi} ⊆ Rd

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SLIDE 89

Further connections – normal surface theory

  • P1, . . . , Pk ⊂ Rd be polytopes (not necessarily full dim)
  • Minkowski sum

P1 + P2 + . . . + Pk = {p1 + p2 + . . . + pk | pi ∈ Pi} ⊆ Rd

+ =

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SLIDE 90

Further connections – normal surface theory

  • P1, . . . , Pk ⊂ Rd be polytopes (not necessarily full dim)
  • Minkowski sum

P1 + P2 + . . . + Pk = {p1 + p2 + . . . + pk | pi ∈ Pi} ⊆ Rd

+ =

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SLIDE 91

Further connections – normal surface theory

  • P1, . . . , Pk ⊂ Rd be polytopes (not necessarily full dim)
  • Minkowski sum

P1 + P2 + . . . + Pk = {p1 + p2 + . . . + pk | pi ∈ Pi} ⊆ Rd

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SLIDE 92

Further connections – normal surface theory

  • Setting: F1, . . . Fd−1 normal (d − 1)-fans in general position

with leafs LFi

1 , LFi 2 , LFi 3

common refinement = collection of rays LF1

i1 ∩ . . . ∩ LFd−1 id−1

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SLIDE 93

Further connections – normal surface theory

  • Setting: F1, . . . Fd−1 normal (d − 1)-fans in general position

with leafs LFi

1 , LFi 2 , LFi 3

common refinement = collection of rays LF1

i1 ∩ . . . ∩ LFd−1 id−1

  • Reformulation: number of rays = number of facets of

Minkowski sum which correspond to a Minkow. sum of facets

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SLIDE 94

Further connections – normal surface theory

  • facets we are interested in

= hitting simplices of the associated colorful Gale transform

  • ⇒ Deza’s bound 1 + d−1

i=1 (|Ci| − 1) becomes 1 + 2d−1

⇒ Burton’s conjecture is true!!

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SLIDE 95

Proof idea

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SLIDE 96

Proof of Main Lemma: Initial configuration

Lemma : βd−1(B, Z2) = 1

  • Let S ∋ 0 be a simplex with vertices v0, v1, . . . , vd.
  • ϕ(Ci) = {vi, −vi, −2vi, −3vi . . . , −
  • |Ci| − 1
  • vi}.
  • B deformation retracts onto the (d − 1)-dimensional sphere,

hence βd−1(B) = 1.

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SLIDE 97

Proof of Main Lemma: Initial configuration

Lemma : βd−1(B, Z2) = 1

  • Let S ∋ 0 be a simplex with vertices v0, v1, . . . , vd.
  • ϕ(Ci) = {vi, −vi, −2vi, −3vi . . . , −
  • |Ci| − 1
  • vi}.
  • B deformation retracts onto the (d − 1)-dimensional sphere,

hence βd−1(B) = 1.

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SLIDE 98

Proof of Main Lemma: Initial configuration

Lemma : βd−1(B, Z2) = 1

  • Let S ∋ 0 be a simplex with vertices v0, v1, . . . , vd.
  • ϕ(Ci) = {vi, −vi, −2vi, −3vi . . . , −
  • |Ci| − 1
  • vi}.
  • B deformation retracts onto the (d − 1)-dimensional sphere,

hence βd−1(B) = 1.

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SLIDE 99

Proof of Main Lemma: Initial configuration

Lemma : βd−1(B, Z2) = 1

  • Let S ∋ 0 be a simplex with vertices v0, v1, . . . , vd.
  • ϕ(Ci) = {vi, −vi, −2vi, −3vi . . . , −
  • |Ci| − 1
  • vi}.
  • B deformation retracts onto the (d − 1)-dimensional sphere,

hence βd−1(B) = 1.

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SLIDE 100

Proof of Main Lemma: Flips

Definition

Let x ∈ Ci be a point.

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SLIDE 101

Proof of Main Lemma: Flips

Definition

Let x ∈ Ci be a point. H is a flipping hyperplane for x if H = aff

  • 0, x0, x1, . . . , xd−2
  • , where xj ∈ Ckj and i = kj = kj′ for

any j = j′.

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SLIDE 102

Proof of Main Lemma: Flips

Definition

Let x ∈ Ci be a point. H is a flipping hyperplane for x if H = aff

  • 0, x0, x1, . . . , xd−2
  • , where xj ∈ Ckj and i = kj = kj′ for

any j = j′.

x

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SLIDE 103

Proof of Main Lemma: Flips

Definition

Let x ∈ Ci be a point. H is a flipping hyperplane for x if H = aff

  • 0, x0, x1, . . . , xd−2
  • , where xj ∈ Ckj and i = kj = kj′ for

any j = j′.

x Definition

A flip (of a colored point x): x x′ s.t. the line segment xx′ crosses at most one flipping hyperplane

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SLIDE 104

Proof of Main Lemma: Flips

Definition

Let x ∈ Ci be a point. H is a flipping hyperplane for x if H = aff

  • 0, x0, x1, . . . , xd−2
  • , where xj ∈ Ckj and i = kj = kj′ for

any j = j′.

x′ Definition

A flip (of a colored point x): x x′ s.t. the line segment xx′ crosses at most one flipping hyperplane

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SLIDE 105

Proof of Main Lemma: Types of flips

Definition

A flip is called

1 safe, if the line segment xx′ does not cross any flipping

hyperplane

x x′ x0

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SLIDE 106

Proof of Main Lemma: Types of flips

Definition

A flip is called

1 safe, if the line segment xx′ does not cross any flipping

hyperplane

2 mild, if the line segment xx′ does cross a flipping hyperplane

aff{0, x0, x1, . . . , xd−2} and 0 / ∈ conv{x, x′, x0, x1, . . . , xd−2}

x x′ x0 x x′ x0

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SLIDE 107

Proof of Main Lemma: Types of flips

Definition

A flip is called

1 safe, if the line segment xx′ does not cross any flipping

hyperplane

2 mild, if the line segment xx′ does cross a flipping hyperplane

aff{0, x0, x1, . . . , xd−2} and 0 / ∈ conv{x, x′, x0, x1, . . . , xd−2}

3 wild, otherwise

x x′ x0 x x′ x0 x x′ x0

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SLIDE 108

Proof of Main Lemma: Safe and mild flips

1 a safe flip preserves B

⇒ it preserves βd−1(B)

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SLIDE 109

Proof of Main Lemma: Safe and mild flips

1 a safe flip preserves B

⇒ it preserves βd−1(B) ⇒ we may assume that all the points are in general position

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SLIDE 110

Proof of Main Lemma: Safe and mild flips

1 a safe flip preserves B

⇒ it preserves βd−1(B) ⇒ we may assume that all the points are in general position

2 a mild flip preserves B

⇒ it preserves βd−1(B)

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SLIDE 111

Proof of Main Lemma: Safe and mild flips

1 a safe flip preserves B

⇒ it preserves βd−1(B) ⇒ we may assume that all the points are in general position

2 a mild flip preserves B

⇒ it preserves βd−1(B)

x x′

slide-112
SLIDE 112

Proof of Main Lemma: Safe and mild flips

1 a safe flip preserves B

⇒ it preserves βd−1(B) ⇒ we may assume that all the points are in general position

2 a mild flip preserves B

⇒ it preserves βd−1(B)

x x′

slide-113
SLIDE 113

Proof of Main Lemma: Safe and mild flips

1 a safe flip preserves B

⇒ it preserves βd−1(B) ⇒ we may assume that all the points are in general position

2 a mild flip preserves B

⇒ it preserves βd−1(B)

x x′

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SLIDE 114

Proof of Main Lemma: Wild flips

Wild flips do change B. B′ = simpl. complex after the flip σ0 a d-simplex present in B′ and not in B

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SLIDE 115

Proof of Main Lemma: Wild flips

Wild flips do change B. B′ = simpl. complex after the flip σ0 a d-simplex present in B′ and not in B

slide-116
SLIDE 116

Proof of Main Lemma: Wild flips

Wild flips do change B. B′ = simpl. complex after the flip σ0 a d-simplex present in B′ and not in B

slide-117
SLIDE 117

Proof of Main Lemma: Wild flips

Wild flips do change B. B′ = simpl. complex after the flip σ0 a d-simplex present in B′ and not in B

x x′

slide-118
SLIDE 118

Proof of Main Lemma: Wild flips

Wild flips do change B. B′ = simpl. complex after the flip σ0 a d-simplex present in B′ and not in B

x x′

slide-119
SLIDE 119

Proof of Main Lemma: Wild flips

Wild flips do change B. B′ = simpl. complex after the flip σ0 a d-simplex present in B′ and not in B

x x′

slide-120
SLIDE 120

Proof of Main Lemma: Wild flips

Wild flips do change B. B′ = simpl. complex after the flip σ0 a d-simplex present in B′ and not in B

x x′

slide-121
SLIDE 121

Proof of Main Lemma: Wild flips

Wild flips do change B. B′ = simpl. complex after the flip σ0 a d-simplex present in B′ and not in B

x x′

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SLIDE 122

Proof of Main Lemma: Wild flips

Wild flips do change B. B′ = simpl. complex after the flip σ0 a d-simplex present in B′ and not in B σ1, . . . , σr all d-simplices that are in B and not in B′

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SLIDE 123

Proof of Main Lemma: Wild flips

Wild flips do change B. B′ = simpl. complex after the flip σ0 a d-simplex present in B′ and not in B σ1, . . . , σr all d-simplices that are in B and not in B′ τ1, . . . , τs all d-simplices present in both B and B′

slide-124
SLIDE 124

Proof of Main Lemma: Wild flips

Wild flips do change B. B′ = simpl. complex after the flip σ0 a d-simplex present in B′ and not in B σ1, . . . , σr all d-simplices that are in B and not in B′ τ1, . . . , τs all d-simplices present in both B and B′ Since βd−1(B) = 1, every (d − 1)-cycle z in B can be expressed as z =

  • i∈I

∂σi +

  • j∈J

∂τj, where I ⊆ {0, 1, . . . , r} and J ⊆ {1, . . . , s}.

slide-125
SLIDE 125

Proof of Main Lemma: Wild flips

Wild flips do change B. B′ = simpl. complex after the flip σ0 a d-simplex present in B′ and not in B σ1, . . . , σr all d-simplices that are in B and not in B′ τ1, . . . , τs all d-simplices present in both B and B′ Since βd−1(B) = 1, every (d − 1)-cycle z in B can be expressed as z =

  • i∈I

∂σi +

  • j∈J

∂τj, where I ⊆ {0, 1, . . . , r} and J ⊆ {1, . . . , s}. ∂τi and ∂σ0 boundaries in B′ ⇒ ∂σ1, . . . , ∂σr generate Hd−1(B′).

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SLIDE 126

Proof of Main Lemma

Clearly ∂σ1 is not zero homologous, therefore βd−1(B′) ≥ 1. Lemma: For every k > 0, the cycle ∂σ1 + ∂σk is contained in a subcomplex C with βd−1(C) = 0.

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SLIDE 127

Proof of Main Lemma

Clearly ∂σ1 is not zero homologous, therefore βd−1(B′) ≥ 1. Lemma: For every k > 0, the cycle ∂σ1 + ∂σk is contained in a subcomplex C with βd−1(C) = 0.

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SLIDE 128

Proof of Main Lemma

Clearly ∂σ1 is not zero homologous, therefore βd−1(B′) ≥ 1. Lemma: For every k > 0, the cycle ∂σ1 + ∂σk is contained in a subcomplex C with βd−1(C) = 0.

x x′

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SLIDE 129

Proof of Main Lemma

Clearly ∂σ1 is not zero homologous, therefore βd−1(B′) ≥ 1. Lemma: For every k > 0, the cycle ∂σ1 + ∂σk is contained in a subcomplex C with βd−1(C) = 0.

x x′

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SLIDE 130

Proof of Main Lemma

Clearly ∂σ1 is not zero homologous, therefore βd−1(B′) ≥ 1. Lemma: For every k > 0, the cycle ∂σ1 + ∂σk is contained in a subcomplex C with βd−1(C) = 0.

x x′

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SLIDE 131

Proof of Main Lemma

Clearly ∂σ1 is not zero homologous, therefore βd−1(B′) ≥ 1. Lemma: For every k > 0, the cycle ∂σ1 + ∂σk is contained in a subcomplex C with βd−1(C) = 0.

x x′

⇒ all (d − 1)-cycles in C are zero homologous

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SLIDE 132

Proof of Main Lemma

Clearly ∂σ1 is not zero homologous, therefore βd−1(B′) ≥ 1. Lemma: For every k > 0, the cycle ∂σ1 + ∂σk is contained in a subcomplex C with βd−1(C) = 0.

x x′

⇒ all (d − 1)-cycles in C are zero homologous ⇒ ∂σ1 and ∂σk are homologous in B′ for all k and βd−1(B′) = 1 as claimed.

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SLIDE 133

Thank you for your attention!