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

Upper bound Theorem (Adiprasito, Brinkmann, Padrol, Pat´ ak, P, Sanyal) d � � � cdepth C ≤ 1 + Card C i − 1 . i =0 • for Card C i = d + 1, we have Deza’s upper bound 1 + d d +1 • the bound is tight!

Upper bound Theorem (Adiprasito, Brinkmann, Padrol, Pat´ ak, P, Sanyal) d � � � cdepth C ≤ 1 + Card C i − 1 . i =0 • for Card C i = d + 1, we have Deza’s upper bound 1 + d d +1 • the bound is tight!

Topological reformulation

Topological approach • A = abstract simpl. complex of all colorful sets in � C i • B = all sets S ⊂ � C i s.t. ϕ ( S ) is non-hitting • f i ( K ) = number of i -dim simplices in K • cdepth( C ) = f d ( A ) − f d ( B ) • A ( d − 1) = B ( d − 1) ⇒ for i < d f i ( A ) = f i ( B ) β i ( A ) = � � ⇒ for i < d − 1 β i ( B )

Topological approach • A = abstract simpl. complex of all colorful sets in � C i • B = all sets S ⊂ � C i s.t. ϕ ( S ) is non-hitting • f i ( K ) = number of i -dim simplices in K • cdepth( C ) = f d ( A ) − f d ( B ) • A ( d − 1) = B ( d − 1) ⇒ for i < d f i ( A ) = f i ( B ) β i ( A ) = � � ⇒ for i < d − 1 β i ( B )

Topological approach • A = abstract simpl. complex of all colorful sets in � C i • B = all sets S ⊂ � C i s.t. ϕ ( S ) is non-hitting • f i ( K ) = number of i -dim simplices in K • cdepth( C ) = f d ( A ) − f d ( B ) • A ( d − 1) = B ( d − 1) ⇒ for i < d f i ( A ) = f i ( B ) β i ( A ) = � � ⇒ for i < d − 1 β i ( B )

Topological approach • A = abstract simpl. complex of all colorful sets in � C i • B = all sets S ⊂ � C i s.t. ϕ ( S ) is non-hitting • f i ( K ) = number of i -dim simplices in K • cdepth( C ) = f d ( A ) − f d ( B ) • A ( d − 1) = B ( d − 1) ⇒ for i < d f i ( A ) = f i ( B ) β i ( A ) = � � ⇒ for i < d − 1 β i ( B )

Topological approach • A = abstract simpl. complex of all colorful sets in � C i • B = all sets S ⊂ � C i s.t. ϕ ( S ) is non-hitting • f i ( K ) = number of i -dim simplices in K • cdepth( C ) = f d ( A ) − f d ( B ) • A ( d − 1) = B ( d − 1) ⇒ for i < d f i ( A ) = f i ( B ) β i ( A ) = � � ⇒ for i < d − 1 β i ( B )

Topological approach • A = abstract simpl. complex of all colorful sets in � C i • B = all sets S ⊂ � C i s.t. ϕ ( S ) is non-hitting • f i ( K ) = number of i -dim simplices in K • cdepth( C ) = f d ( A ) − f d ( B ) • A ( d − 1) = B ( d − 1) ⇒ for i < d f i ( A ) = f i ( B ) β i ( A ) = � � ⇒ for i < d − 1 β i ( B )

Topological approach • A = abstract simpl. complex of all colorful sets in � C i • B = all sets S ⊂ � C i s.t. ϕ ( S ) is non-hitting • f i ( K ) = number of i -dim simplices in K • cdepth( C ) = f d ( A ) − f d ( B ) • A ( d − 1) = B ( d − 1) ⇒ for i < d f i ( A ) = f i ( B ) β i ( A ) = � � ⇒ for i < d − 1 β i ( B )

Topological approach cdepth C = f d ( A ) − f d ( B )

Topological approach cdepth C = f d ( A ) − f d ( B )

Topological approach cdepth C = f d ( A ) − f d ( B ) � d � d ( − 1) i � ( − 1) i f i ( A ) = χ ( A ) = − 1 + � β i ( A ) i =0 i =0

Topological approach cdepth C = f d ( A ) − f d ( B ) � d � d ( − 1) i � ( − 1) i f i ( A ) = χ ( A ) = − 1 + � β i ( A ) i =0 i =0 � d � � d − 1 � ( − 1) i � ⇒ f d ( A ) = ( − 1) d ( − 1) i f i ( A ) β i ( A ) + 1 − i =0 i =0

Topological approach cdepth C = ( − 1) d � d � � d − 1 � ( − 1) i � ( − 1) i f i ( A ) β i ( A ) + 1 − − f d ( B ) i =0 i =0 � d � � d − 1 � ( − 1) i � ⇒ f d ( A ) = ( − 1) d ( − 1) i f i ( A ) β i ( A ) + 1 − i =0 i =0

Topological approach cdepth C = ( − 1) d � d � � d − 1 � ( − 1) i � ( − 1) i f i ( A ) β i ( A ) + 1 − − f d ( B ) i =0 i =0

Topological approach cdepth C = ( − 1) d � d � � d − 1 � ( − 1) i � ( − 1) i f i ( A ) β i ( A ) + 1 − − f d ( B ) i =0 i =0 � d � � d − 1 � ( − 1) i � f d ( B ) = ( − 1) d ( − 1) i f i ( B ) β i ( B ) + 1 − i =0 i =0

Topological approach cdepth C = ( − 1) d � d � d − 1 � ( − 1) i � ( − 1) i f i ( A ) β i ( A ) + 1 − i =0 i =0 � � d − 1 � d ( − 1) i � ( − 1) i f i ( B ) − β i ( B ) − 1 + i =0 i =0 � d � � d − 1 � ( − 1) i � f d ( B ) = ( − 1) d ( − 1) i f i ( B ) β i ( B ) + 1 − i =0 i =0

Topological approach cdepth C = ( − 1) d � d � d − 1 � ( − 1) i � ( − 1) i f i ( A ) β i ( A ) + 1 − i =0 i =0 � � d − 1 � d ( − 1) i � ( − 1) i f i ( B ) − β i ( B ) − 1 + i =0 i =0

Topological approach cdepth C = ( − 1) d � d � d − 1 � ( − 1) i � ( − 1) i f i ( A ) β i ( A ) − i =0 i =0 � � d − 1 � d ( − 1) i � ( − 1) i f i ( B ) − β i ( B )+ i =0 i =0

Topological approach cdepth C = ( − 1) d � d � d − 1 � ( − 1) i � ( − 1) i f i ( A ) β i ( A ) − i =0 i =0 � � d − 1 � d ( − 1) i � ( − 1) i f i ( B ) − β i ( B )+ i =0 i =0

Topological approach cdepth C = ( − 1) d � d � d − 1 � ( − 1) i � ( − 1) i f i ( A ) β i ( A ) − i =0 i =0 � � d − 1 � d ( − 1) i � ( − 1) i f i ( B ) − β i ( B )+ i =0 i =0 For i < d : f i ( A ) = f i ( B )

Topological approach cdepth C = ( − 1) d � d � ( − 1) i � β i ( A ) i =0 � � d ( − 1) i � − β i ( B ) i =0

Topological approach cdepth C = ( − 1) d � d � ( − 1) i � β i ( A ) i =0 � � d ( − 1) i � − β i ( B ) i =0

Topological approach cdepth C = ( − 1) d � d � ( − 1) i � β i ( A ) i =0 � � d ( − 1) i � − β i ( B ) i =0 For i < d − 1: � β i ( A ) = � β i ( B )

Topological approach cdepth C = ( − 1) d � ( − 1) d � β d ( A ) + ( − 1) d − 1 � β d − 1 ( A ) � − ( − 1) d � β d ( B ) − ( − 1) d − 1 � β d − 1 ( B )

Topological approach cdepth C = ( − 1) d � ( − 1) d � β d ( A ) + ( − 1) d − 1 � β d − 1 ( A ) � − ( − 1) d � β d ( B ) − ( − 1) d − 1 � β d − 1 ( B )

Topological approach cdepth C = � β d ( A ) − � β d − 1 ( A ) − � β d ( B ) + � β d − 1 ( B )

Topological approach cdepth C = � β d ( A ) − � β d − 1 ( A ) − � β d ( B ) + � β d − 1 ( B )

Topological approach cdepth C = � β d ( A ) − 0 − � β d ( B ) + � β d − 1 ( B )

Topological approach cdepth C = � β d ( A ) − � β d ( B ) + � β d − 1 ( B )

Topological approach cdepth C = � β d ( A ) − � β d ( B ) + � β d − 1 ( B )

Topological approach � � � d − � β d ( B ) + � cdepth C = | C i | − 1 β d − 1 ( B ) i =0

Topological approach � � � d − � β d ( B ) + � cdepth C = | C i | − 1 β d − 1 ( B ) i =0

Topological approach � � � d − � β d ( B ) + � cdepth C = | C i | − 1 β d − 1 ( B ) i =0 Our main Lemma: � β d − 1 ( B ) = 1

Topological approach � � � d − � cdepth C = | C i | − 1 β d ( B ) + 1 i =0

Topological approach � � � d − � cdepth C = | C i | − 1 β d ( B ) + 1 i =0 � � � d ⇒ cdepth C ≤ | C i | − 1 + 1 i =0

Main lemma Lemma � β d − 1 ( B ; Z 2 ) = 1 . Proof idea: 1 First show for a special configuration of points: 2 Use flips preserving � β d − 1 ( B ; Z 2 )

Main lemma Lemma � β d − 1 ( B ; Z 2 ) = 1 . Proof idea: 1 First show for a special configuration of points: 2 Use flips preserving � β d − 1 ( B ; Z 2 )

Main lemma Lemma � β d − 1 ( B ; Z 2 ) = 1 . Proof idea: 1 First show for a special configuration of points: 0 2 Use flips preserving � β d − 1 ( B ; Z 2 )

Main lemma Lemma � β d − 1 ( B ; Z 2 ) = 1 . Proof idea: 1 First show for a special configuration of points: 0 2 Use flips preserving � β d − 1 ( B ; Z 2 )

Further connections

Further connections – normal surface theory • normal d -fan = collection of polyhedral cones

Further connections – normal surface theory • normal d -fan = collection of polyhedral cones 1-fan, given by normals of a triangle

Further connections – normal surface theory • normal d -fan = collection of polyhedral cones 1-fan, given by normals of a triangle

Further connections – normal surface theory • normal d -fan = collection of polyhedral cones 2-fan; halfplanes = leafs

Further connections – normal surface theory • normal d -fan = collection of polyhedral cones 2-fan; halfplanes = leafs • two (and more) normal d -fans ⇒ common refinement

Further connections – normal surface theory • Setting: F 1 , . . . F d − 1 normal ( d − 1)-fans in general position with leafs L F i 1 , L F i 2 , L F i 3 i 1 ∩ . . . ∩ L F d − 1 common refinement = collection of rays L F 1 i d − 1 • Question: Max number of rays in the common refinement? • Conjecture (Burton’03): 1 + 2 d − 1

Further connections – normal surface theory • Setting: F 1 , . . . F d − 1 normal ( d − 1)-fans in general position with leafs L F i 1 , L F i 2 , L F i 3 i 1 ∩ . . . ∩ L F d − 1 common refinement = collection of rays L F 1 i d − 1 • Question: Max number of rays in the common refinement? • Conjecture (Burton’03): 1 + 2 d − 1

Further connections – normal surface theory • Setting: F 1 , . . . F d − 1 normal ( d − 1)-fans in general position with leafs L F i 1 , L F i 2 , L F i 3 i 1 ∩ . . . ∩ L F d − 1 common refinement = collection of rays L F 1 i d − 1 • Question: Max number of rays in the common refinement? • Conjecture (Burton’03): 1 + 2 d − 1

Further connections – normal surface theory • P 1 , . . . , P k ⊂ R d be polytopes (not necessarily full dim) • Minkowski sum P 1 + P 2 + . . . + P k = { p 1 + p 2 + . . . + p k | p i ∈ P i } ⊆ R d

Further connections – normal surface theory • P 1 , . . . , P k ⊂ R d be polytopes (not necessarily full dim) • Minkowski sum P 1 + P 2 + . . . + P k = { p 1 + p 2 + . . . + p k | p i ∈ P i } ⊆ R d + =

Further connections – normal surface theory • P 1 , . . . , P k ⊂ R d be polytopes (not necessarily full dim) • Minkowski sum P 1 + P 2 + . . . + P k = { p 1 + p 2 + . . . + p k | p i ∈ P i } ⊆ R d + =

Further connections – normal surface theory • P 1 , . . . , P k ⊂ R d be polytopes (not necessarily full dim) • Minkowski sum P 1 + P 2 + . . . + P k = { p 1 + p 2 + . . . + p k | p i ∈ P i } ⊆ R d

Further connections – normal surface theory • Setting: F 1 , . . . F d − 1 normal ( d − 1)-fans in general position with leafs L F i 1 , L F i 2 , L F i 3 i 1 ∩ . . . ∩ L F d − 1 common refinement = collection of rays L F 1 i d − 1

Further connections – normal surface theory • Setting: F 1 , . . . F d − 1 normal ( d − 1)-fans in general position with leafs L F i 1 , L F i 2 , L F i 3 i 1 ∩ . . . ∩ L F d − 1 common refinement = collection of rays L F 1 i d − 1 • Reformulation: number of rays = number of facets of Minkowski sum which correspond to a Minkow. sum of facets

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 ( | C i | − 1) becomes 1 + 2 d − 1 ⇒ Burton’s conjecture is true!!

Proof idea

Proof of Main Lemma: Initial configuration Lemma : � β d − 1 ( B , Z 2 ) = 1 • Let S ∋ 0 be a simplex with vertices v 0 , v 1 , . . . , v d . � � • ϕ ( C i ) = { v i , − v i , − 2 v i , − 3 v i . . . , − | C i | − 1 v i } . • B deformation retracts onto the ( d − 1)-dimensional sphere, hence � β d − 1 ( B ) = 1 .

Proof of Main Lemma: Initial configuration Lemma : � β d − 1 ( B , Z 2 ) = 1 • Let S ∋ 0 be a simplex with vertices v 0 , v 1 , . . . , v d . � � • ϕ ( C i ) = { v i , − v i , − 2 v i , − 3 v i . . . , − | C i | − 1 v i } . • B deformation retracts onto the ( d − 1)-dimensional sphere, hence � β d − 1 ( B ) = 1 .

Proof of Main Lemma: Initial configuration Lemma : � β d − 1 ( B , Z 2 ) = 1 • Let S ∋ 0 be a simplex with vertices v 0 , v 1 , . . . , v d . � � • ϕ ( C i ) = { v i , − v i , − 2 v i , − 3 v i . . . , − | C i | − 1 v i } . 0 • B deformation retracts onto the ( d − 1)-dimensional sphere, hence � β d − 1 ( B ) = 1 .

Proof of Main Lemma: Initial configuration Lemma : � β d − 1 ( B , Z 2 ) = 1 • Let S ∋ 0 be a simplex with vertices v 0 , v 1 , . . . , v d . � � • ϕ ( C i ) = { v i , − v i , − 2 v i , − 3 v i . . . , − | C i | − 1 v i } . 0 • B deformation retracts onto the ( d − 1)-dimensional sphere, hence � β d − 1 ( B ) = 1 .

Proof of Main Lemma: Flips Definition Let x ∈ C i be a point.