Cutting a part from many measures Nevena Pali 6th BMS Student - PowerPoint PPT Presentation

Cutting a part from many measures Nevena Palić 6th BMS Student Conference Nevena Palić Cutting a part from many measures 23.02.2018 1 / 16

Equivariant Topological Combinatorics

Equivariant Topological Combinatorics

Equivariant Topological Combinatorics

Equivariant Topological Combinatorics

Equivariant Topological Combinatorics

Equivariant Topological G Combinatorics G G G G

Equivariant Topological Combinatorics G G G

Equivariant Topological Combinatorics G G G

First example Theorem (Ham Sandwich theorem, Banach 1938) Any collection of d finite absolutely continuous measures in R d can be simultaneously cut into two parts by one hyperplane cut in such a way that each part captures exactly one half of each measure. Figure: from Curiosa Mathematica Nevena Palić Cutting a part from many measures 23.02.2018 2 / 16

First example H H + H − R d Nevena Palić Cutting a part from many measures 23.02.2018 3 / 16

First example N R d H H H + H − R d +1 S R d Nevena Palić Cutting a part from many measures 23.02.2018 3 / 16

First example Proof. S d \ { N, S } R d − → ( µ 1 ( H + p ) , . . . , µ d ( H + p ) − µ 1 ( H − p ) − µ d ( H − �− → p p )) Nevena Palić Cutting a part from many measures 23.02.2018 4 / 16

First example Proof. S d R d − → ( µ 1 ( H + p ) , . . . , µ d ( H + p ) − µ 1 ( H − p ) − µ d ( H − �− → p p )) Nevena Palić Cutting a part from many measures 23.02.2018 4 / 16

First example Proof. S d R d − → ( µ 1 ( H + p ) , . . . , µ d ( H + p ) − µ 1 ( H − p ) − µ d ( H − �− → p p )) Assume that there is no solution, i.e., for every hyperplane H there is some 1 ≤ i ≤ d such that µ i ( H + ) � = µ i ( H − ) . Nevena Palić Cutting a part from many measures 23.02.2018 4 / 16

First example Proof. S d R d − → ( µ 1 ( H + p ) , . . . , µ d ( H + p ) − µ 1 ( H − p ) − µ d ( H − �− → p p )) Assume that there is no solution, i.e., for every hyperplane H there is some 1 ≤ i ≤ d such that µ i ( H + ) � = µ i ( H − ) . S d − → R d \ { 0 } − → S d − 1 Nevena Palić Cutting a part from many measures 23.02.2018 4 / 16

First example Proof. S d R d − → ( µ 1 ( H + p ) , . . . , µ d ( H + p ) − µ 1 ( H − p ) − µ d ( H − �− → p p )) Assume that there is no solution, i.e., for every hyperplane H there is some 1 ≤ i ≤ d such that µ i ( H + ) � = µ i ( H − ) . S d − → R d \ { 0 } − → S d − 1 Group action! S d − → Z 2 S d − 1 Nevena Palić Cutting a part from many measures 23.02.2018 4 / 16

First example Let G be a group that acts on topological spaces X and Y . A map f : X − → Y is G-equivariant if f ( g · x ) = g · f ( x ) for every x ∈ X and every g ∈ G . Nevena Palić Cutting a part from many measures 23.02.2018 5 / 16

First example Let G be a group that acts on topological spaces X and Y . A map f : X − → Y is G-equivariant if f ( g · x ) = g · f ( x ) for every x ∈ X and every g ∈ G . Theorem (Borsuk-Ulam) There is no Z 2 -equivariant map S d → S d − 1 . Nevena Palić Cutting a part from many measures 23.02.2018 5 / 16

First example – discrete version Theorem (Discrete Ham Sandwich theorem, Matoušek) Let A 1 , . . . , A d ⊂ R d be disjoint finite point sets in general position. Then there exists a hyperplane H that bisects each set A i , such that there are exactly ⌊ 1 2 | A i |⌋ points from the set A i in each of the open half-spaces defined by H , for every 1 ≤ i ≤ d . Nevena Palić Cutting a part from many measures 23.02.2018 6 / 16

First example – discrete version Theorem (Discrete Ham Sandwich theorem, Matoušek) Let A 1 , . . . , A d ⊂ R d be disjoint finite point sets in general position. Then there exists a hyperplane H that bisects each set A i , such that there are exactly ⌊ 1 2 | A i |⌋ points from the set A i in each of the open half-spaces defined by H , for every 1 ≤ i ≤ d . Nevena Palić Cutting a part from many measures 23.02.2018 6 / 16

Partitions and measures Definition Let d ≥ 1 and n ≥ 1 be integers. An ordered collection of closed subsets ( C 1 , . . . , C n ) of R d is called a partition of R d if (1) � n i =1 C i = R d , (2) int( C i ) � = ∅ for every 1 ≤ i ≤ n , and (3) int( C i ) ∩ int( C j ) = ∅ for all 1 ≤ i < j ≤ n . A partition ( C 1 , . . . , C n ) is called convex if all subsets C 1 , . . . , C n are convex. Nevena Palić Cutting a part from many measures 23.02.2018 7 / 16

Partitions and measures Definition Let d ≥ 1 and n ≥ 1 be integers. An ordered collection of closed subsets ( C 1 , . . . , C n ) of R d is called a partition of R d if (1) � n i =1 C i = R d , (2) int( C i ) � = ∅ for every 1 ≤ i ≤ n , and (3) int( C i ) ∩ int( C j ) = ∅ for all 1 ≤ i < j ≤ n . A partition ( C 1 , . . . , C n ) is called convex if all subsets C 1 , . . . , C n are convex. We consider finite absolutely continuous Borel measures. Nevena Palić Cutting a part from many measures 23.02.2018 7 / 16

Motivation Conjecture (Holmsen, Kynčl, Valculescu, 2017) d ≥ 2 , ℓ ≥ 2 , m ≥ 2 and n ≥ 1 integers, m ≥ d and ℓ ≥ d , X ⊆ R d , | X | = ℓn , X in general position, X is colored with at least m different colors. a partition of X into n subsets of size ℓ such that each subset contains points colored by at least d colors. Nevena Palić Cutting a part from many measures 23.02.2018 8 / 16

Motivation Conjecture (Holmsen, Kynčl, Valculescu, 2017) d ≥ 2 , ℓ ≥ 2 , m ≥ 2 and n ≥ 1 integers, m ≥ d and ℓ ≥ d , X ⊆ R d , | X | = ℓn , X in general position, X is colored with at least m different colors. a partition of X into n subsets of size ℓ such that each subset contains points colored by at least d colors. Then there exists such a partition of X that in addition has the property that the convex hulls of the n subsets are pairwise disjoint. Nevena Palić Cutting a part from many measures 23.02.2018 8 / 16

Motivation d = 2 , m = 3 , n = 6 , l = 3 Nevena Palić Cutting a part from many measures 23.02.2018 9 / 16

Motivation d = 2 , m = 3 , n = 6 , l = 3 Nevena Palić Cutting a part from many measures 23.02.2018 9 / 16

Cutting a part from many measures Theorem (Blagojević, P., Ziegler, 2017) d ≥ 2 , m ≥ 2 , and c ≥ 2 integers, n = p k a prime power, m ≥ n ( c − d ) + dn p − n p + 1 , µ 1 , . . . , µ m positive finite absolutely continuous measures on R d . Nevena Palić Cutting a part from many measures 23.02.2018 10 / 16

Cutting a part from many measures Theorem (Blagojević, P., Ziegler, 2017) d ≥ 2 , m ≥ 2 , and c ≥ 2 integers, n = p k a prime power, m ≥ n ( c − d ) + dn p − n p + 1 , µ 1 , . . . , µ m positive finite absolutely continuous measures on R d . Then there exists a convex partition ( C 1 , . . . , C n ) such that Nevena Palić Cutting a part from many measures 23.02.2018 10 / 16

Cutting a part from many measures Theorem (Blagojević, P., Ziegler, 2017) d ≥ 2 , m ≥ 2 , and c ≥ 2 integers, n = p k a prime power, m ≥ n ( c − d ) + dn p − n p + 1 , µ 1 , . . . , µ m positive finite absolutely continuous measures on R d . Then there exists a convex partition ( C 1 , . . . , C n ) such that # { j : 1 ≤ j ≤ m, µ j ( C i ) > 0 } ≥ c and µ m ( C 1 ) = · · · = µ m ( C n ) = 1 nµ m ( R d ) , for every 1 ≤ i ≤ n . Nevena Palić Cutting a part from many measures 23.02.2018 10 / 16

Cutting a part from many measures Example n = 5 c = 4 d = 3 m = 8 Nevena Palić Cutting a part from many measures 23.02.2018 11 / 16

CS/TM scheme EMP( µ m , n ) – all convex partitions of R d into n convex pieces that equipart the measure µ m Nevena Palić Cutting a part from many measures 23.02.2018 12 / 16

CS/TM scheme EMP( µ m , n ) – all convex partitions of R d into n convex pieces that equipart the measure µ m R ( m − 1) × n ∼ = ( R m − 1 ) n f : EMP( µ m , n ) − →   µ 1 ( C 1 ) µ 1 ( C 2 ) . . . µ 1 ( C n ) µ 2 ( C 1 ) µ 2 ( C 2 ) . . . µ 2 ( C n )   ( C 1 , . . . , C n ) �− →  . . .  ... . . .   . . .   µ m − 1 ( C 1 ) µ m − 1 ( C 2 ) µ m − 1 ( C n ) . . . Nevena Palić Cutting a part from many measures 23.02.2018 12 / 16

CS/TM scheme EMP( µ m , n ) – all convex partitions of R d into n convex pieces that equipart the measure µ m R ( m − 1) × n ∼ = ( R m − 1 ) n f : EMP( µ m , n ) − →   µ 1 ( C 1 ) µ 1 ( C 2 ) . . . µ 1 ( C n ) µ 2 ( C 1 ) µ 2 ( C 2 ) . . . µ 2 ( C n )   ( C 1 , . . . , C n ) �− →  . . .  ... . . .   . . .   µ m − 1 ( C 1 ) µ m − 1 ( C 2 ) µ m − 1 ( C n ) . . . f is S n -equivariant. Nevena Palić Cutting a part from many measures 23.02.2018 12 / 16

CS/TM scheme n � ( y jk ) ∈ R ( m − 1) × n : � � y jk = µ j ( R d ) for every 1 ≤ j ≤ m − 1 V = k =1 ∼ R ( m − 1) × ( n − 1) = Nevena Palić Cutting a part from many measures 23.02.2018 13 / 16