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Discrete geometry - Personal reflections on someworks by Jirı Matousek
Gil Kalai
June 23, 2015
LFT 100 meeting, Budapest 2015.
Gil Kalai Discrete geometry and Jirka Matousek
Some general themes for this lecture
I Graphs and hypergraphs arising in geometry are very special
I Phenomena in discrete geometry often have have strongtopological flavour.
I Phenomena in discrete geometry often have very generalcombinatorial underlying explanation.
Gil Kalai Discrete geometry and Jirka Matousek
Some general themes for this lecture
I Graphs and hypergraphs arising in geometry are very special
I Phenomena in discrete geometry often have have strongtopological flavour.
I Phenomena in discrete geometry often have very generalcombinatorial underlying explanation.
Gil Kalai Discrete geometry and Jirka Matousek
Some general themes for this lecture
I Graphs and hypergraphs arising in geometry are very special
I Phenomena in discrete geometry often have have strongtopological flavour.
I Phenomena in discrete geometry often have very generalcombinatorial underlying explanation.
Gil Kalai Discrete geometry and Jirka Matousek
Gil Kalai Discrete geometry and Jirka Matousek
A new result about approximations of smooth convex bodies bypolytopes
It is known that if a simplicial convex polytope P ε-approximates aC 2-convex body K . Then, the number of vertices of P isΩ(ε−(d−1)/2).Theorem: Adiprasito, Nevo and Samper
gk(P) = Ω(ε−(d−1)/2).
This proves a conjecture I made in the 90s.
Gil Kalai Discrete geometry and Jirka Matousek
1: Linear Programming
Gil Kalai Discrete geometry and Jirka Matousek
A randomized simplex algorihms: Random Facet
RandomFacet (Sharir and Welzl)
I Start from a vertex v and choose a random facet containing it
I Apply the algorithm recursively inside this facet
I Repeat!
Gil Kalai Discrete geometry and Jirka Matousek
Ranom Facet is subexponential!
Theorem (Matousek, Sharir, Welzl and Kalai, 1992): RandomFacet requires a subexponential expected running time for everyLP problem with d variables and n ineqialities.
Expected number of pivot steps e√
K log dn.
Gil Kalai Discrete geometry and Jirka Matousek
A crucial fact in analysis
Let a1 < a2 < a3 . . . be a monotone sequence of reals:
If an+1 − an = Average(a1, a2 . . . , an) then an = exp(K√
n).
If an+1 − an = Median(a1, a2 . . . , an) then an = exp(K log2 n).
I also discovered that nlog d = d log n.
Gil Kalai Discrete geometry and Jirka Matousek
A crucial fact in analysis
Let a1 < a2 < a3 . . . be a monotone sequence of reals:
If an+1 − an = Average(a1, a2 . . . , an) then an = exp(K√
n).
If an+1 − an = Median(a1, a2 . . . , an) then an = exp(K log2 n).
I also discovered that nlog d = d log n.
Gil Kalai Discrete geometry and Jirka Matousek
Abstraction of LP
1. Abstract objective function on polytopes: unique sink (localmaximum) on every face
2. “Polytopes” can be replaced by more abstract objects as well.
Gil Kalai Discrete geometry and Jirka Matousek
Random Facet can be exponential (in√
d) for abstract cubes
This is an early result from 1994 by Jiri Matousek.
Gil Kalai Discrete geometry and Jirka Matousek
Random Edge
I Start from a vertex v and choose a random edge containing it
I Move to the other vertex if this improves matters
I Repeat!
Gil Kalai Discrete geometry and Jirka Matousek
Random edge can be exponential(*) for abstract cubes
This is a result by Jiri Matousek and Tibor Szabo from 2004.
Gil Kalai Discrete geometry and Jirka Matousek
Recent breakthrough: Random edge and random facet canbe exponential(*) for LP
This is a 2010 breakthrough by Oliver Friedmann, Thomas Hansen,and Uri Zwick
Gil Kalai Discrete geometry and Jirka Matousek
Challenges
Better pivot rules:What about RandomFace algorithm?What about random-walk based algorithms?
Unique sink orientations
Diameter of polytopes and “abstract polytopes” (the polynomialHirsch conjecture)
Can Geometry help?
Average (and smoothed) case of randomized pivot rules
Gil Kalai Discrete geometry and Jirka Matousek
Part 2: Our art gallery theorem
Gil Kalai Discrete geometry and Jirka Matousek
Theorem (Kalai and Matousek 1997: For a simply connectedplanar gallery of area 1, if a guard in every location sees points ofarea ≥ ε then Cε log(1/ε) guards suffices.
Key: Bounded VC dimension and ε-nets.
Question: Can we get rid of log(1/ε)
Major general question: For bounded VC dimension when canwe get read of the log(1/ε)(related to many things, e.g., to a famous conjecture by Danzer.)
Gil Kalai Discrete geometry and Jirka Matousek
Theorem (Kalai and Matousek 1997: For a simply connectedplanar gallery of area 1, if a guard in every location sees points ofarea ≥ ε then Cε log(1/ε) guards suffices.
Key: Bounded VC dimension and ε-nets.
Question: Can we get rid of log(1/ε)
Major general question: For bounded VC dimension when canwe get read of the log(1/ε)(related to many things, e.g., to a famous conjecture by Danzer.)
Gil Kalai Discrete geometry and Jirka Matousek
Theorem (Kalai and Matousek 1997: For a simply connectedplanar gallery of area 1, if a guard in every location sees points ofarea ≥ ε then Cε log(1/ε) guards suffices.
Key: Bounded VC dimension and ε-nets.
Question: Can we get rid of log(1/ε)
Major general question: For bounded VC dimension when canwe get read of the log(1/ε)(related to many things, e.g., to a famous conjecture by Danzer.)
Gil Kalai Discrete geometry and Jirka Matousek
Gil Kalai Discrete geometry and Jirka Matousek
Part 3: Helly theorem, and the fractional Helly theorem
Gil Kalai Discrete geometry and Jirka Matousek
Helly numbers and Helly’s theorem
A family F of sets has Helly number k if for every finite subfamilyG ⊂ F , |G| ≥ k, if every k members of G have a point in common,then all members of G have a point in common.And, moreover, k is the smallest integer with this property.
Helly’s theorem:The family of compact convex sets in Rd has Helly number d + 1.
Gil Kalai Discrete geometry and Jirka Matousek
Helly orders
A family F has Helly order k if for every finite subfamily G,|G| ≥ k, with the property that all intersections of sets in G is inF , if every k members of G have a point in common, then allmembers of G have a point in common.And, moreover, k is the smallest integer with this property.
Gil Kalai Discrete geometry and Jirka Matousek
Topological Helly’s theorem
Topological Helly’s theorem: (proved by Helly himself!) The classof compact sets homehomorphic to a ball in Rd have Helly orderd + 1.
Gil Kalai Discrete geometry and Jirka Matousek
Helly orders for sets with bounded complexity
For a compact set K in Rd let b(K ) be the minimal number suchthat K can be presented as the union of b(K ) compact convexsets. Let b0(K ) be the minimum number so that K can bepresented as the union of disjoint convex sets.
Theorem (Matousek and Alon and Kalai (around 1995)) :The class of compact sets K in Rd with b(K ) ≤ b have boundedHelly order.
Theorem (Amenta (following Motzkin-Grunbaum, Larmanand Morris)) : The class of compact sets K in Rd withb0(K ) ≤ b have Helly order b(d + 1).
Curious question: If n > d + 1 and X1, . . . Xn compact sets in Rd
such that every j , j < n the intersection of every j sets is the unionof two closed nonempty convex sets, is there always a point incommon to all sets?
Gil Kalai Discrete geometry and Jirka Matousek
Helly orders for sets with bounded complexity
For a compact set K in Rd let b(K ) be the minimal number suchthat K can be presented as the union of b(K ) compact convexsets. Let b0(K ) be the minimum number so that K can bepresented as the union of disjoint convex sets.
Theorem (Matousek and Alon and Kalai (around 1995)) :The class of compact sets K in Rd with b(K ) ≤ b have boundedHelly order.
Theorem (Amenta (following Motzkin-Grunbaum, Larmanand Morris)) : The class of compact sets K in Rd withb0(K ) ≤ b have Helly order b(d + 1).
Curious question: If n > d + 1 and X1, . . . Xn compact sets in Rd
such that every j , j < n the intersection of every j sets is the unionof two closed nonempty convex sets, is there always a point incommon to all sets?
Gil Kalai Discrete geometry and Jirka Matousek
Helly orders for sets with bounded complexity
For a compact set K in Rd let b(K ) be the minimal number suchthat K can be presented as the union of b(K ) compact convexsets. Let b0(K ) be the minimum number so that K can bepresented as the union of disjoint convex sets.
Theorem (Matousek and Alon and Kalai (around 1995)) :The class of compact sets K in Rd with b(K ) ≤ b have boundedHelly order.
Theorem (Amenta (following Motzkin-Grunbaum, Larmanand Morris)) : The class of compact sets K in Rd withb0(K ) ≤ b have Helly order b(d + 1).
Curious question: If n > d + 1 and X1, . . . Xn compact sets in Rd
such that every j , j < n the intersection of every j sets is the unionof two closed nonempty convex sets, is there always a point incommon to all sets?
Gil Kalai Discrete geometry and Jirka Matousek
The fractional Helly property
Let F be a family of sets. F satisfies The fractional Helly property(FHP) with index k, if for every α there is β such that for everysubfamily G of n sets if a fraction α of all k-subfamilies areintersecting then a fraction β of all members of G have nonemptyintersection.
The strong FHP with index k: Also α → 1 when β → 1.
Gil Kalai Discrete geometry and Jirka Matousek
The piercing property
Piercing property with index k: For every p > k there is f (p) suchthat if from every p sets, k sets have a point in common thenthere are f (p) points such that every set contains one of them.
Gil Kalai Discrete geometry and Jirka Matousek
Theorem (Katchalski and Liu, Eckhoff, Kalai) around 1980:Convex sets in Rd have the strong fractional Helly property withindex d + 1.
Theorem (Alon and Kleitman, 1992): Convex sets in Rd havethe piercing property with index d + 1.
Theorem (Alon, Kalai, Matousek, Meshulam, 2002):The fractional Helly property implies piercing property with thesame parameter.
Gil Kalai Discrete geometry and Jirka Matousek
The Barany-Matousek theorem
Integral Helly theorem (Scarf): Let F be a collection of nconvex sets in Rd . If every 2d sets in F have an integer point incommon then there is an integer point common to all of the sets.
In other words: ”integral convex sets” in Rd have Helly number 2d .
Barany-Matousek fractional Helly Theorem:Integral convex sets in Rd satisfy the fractional Helly property withparameter d+1.
In particular:There is a positive constant α(d) such that the followingstatement holds:Let F be a collection of n convex sets in Rd . If every d + 1 sets inF have an integer point in common, then there is an integer pointcommon to α(d)n of the sets.
Gil Kalai Discrete geometry and Jirka Matousek
The Barany-Matousek theorem
Integral Helly theorem (Scarf): Let F be a collection of nconvex sets in Rd . If every 2d sets in F have an integer point incommon then there is an integer point common to all of the sets.
In other words: ”integral convex sets” in Rd have Helly number 2d .
Barany-Matousek fractional Helly Theorem:Integral convex sets in Rd satisfy the fractional Helly property withparameter d+1.
In particular:There is a positive constant α(d) such that the followingstatement holds:Let F be a collection of n convex sets in Rd . If every d + 1 sets inF have an integer point in common, then there is an integer pointcommon to α(d)n of the sets.
Gil Kalai Discrete geometry and Jirka Matousek
What type of properties implies fractional Helly?
Theorem: (Matousek) Bounded VC-dimension implies thefractional Helly property.
This inspired the following:
Conjecture (Kalai and Meshulam):Fractional Helly of parameter k follows from polynomial growth(like nk) of the total Betti numbers of the nerve.
Gil Kalai Discrete geometry and Jirka Matousek
The case k = 0
For a graph G , I (G ) is the independent complex of G and β(I (G ))is the sum of (reduced) Betti numbers of I (H).
Conjecture: Let G be a graph. If βI (H) < K for every inducedsubgraph then χ(G ) is bounded.
What about K=1.Conjecture: β(I (H)) ≤ 1 for every induced subgraph H iff G doesnot contain an induced cycle of length 0(mod 3).
Gyarfas type question (Kalai and Meshulam): Is there auniform upper bound for the chromatic number of all graphs Gsuch that all induced cycles in G are of length 1 or 2 modulo 3?Answer: Yes! Theorem by Bonamy, Charbitz and Thomasse.
Gil Kalai Discrete geometry and Jirka Matousek
The case k = 0
For a graph G , I (G ) is the independent complex of G and β(I (G ))is the sum of (reduced) Betti numbers of I (H).
Conjecture: Let G be a graph. If βI (H) < K for every inducedsubgraph then χ(G ) is bounded.What about K=1.Conjecture: β(I (H)) ≤ 1 for every induced subgraph H iff G doesnot contain an induced cycle of length 0(mod 3).
Gyarfas type question (Kalai and Meshulam): Is there auniform upper bound for the chromatic number of all graphs Gsuch that all induced cycles in G are of length 1 or 2 modulo 3?Answer: Yes! Theorem by Bonamy, Charbitz and Thomasse.
Gil Kalai Discrete geometry and Jirka Matousek
The case k = 0
For a graph G , I (G ) is the independent complex of G and β(I (G ))is the sum of (reduced) Betti numbers of I (H).
Conjecture: Let G be a graph. If βI (H) < K for every inducedsubgraph then χ(G ) is bounded.What about K=1.Conjecture: β(I (H)) ≤ 1 for every induced subgraph H iff G doesnot contain an induced cycle of length 0(mod 3).
Gyarfas type question (Kalai and Meshulam): Is there auniform upper bound for the chromatic number of all graphs Gsuch that all induced cycles in G are of length 1 or 2 modulo 3?Answer: Yes! Theorem by Bonamy, Charbitz and Thomasse.
Gil Kalai Discrete geometry and Jirka Matousek
Part 4: Topological methods (mainly a la Borsuk)
Gil Kalai Discrete geometry and Jirka Matousek
Tverberg’s theorem
Tverberg’s theorem: Every set of points x1, x2, . . . , xm form = (d + 1)(r − 1) + 1 can be divided into m pairwise disjointparts X1,X2, . . . ,Xr such that
conv(X1) ∩ conv(X2),∩ · · · ∩ conv(Xr ) 6= ∅.
History: Birch (conjectured), Rado (proved a weaker result),Tverberg (proved), Tverberg (reproved), Tverberg & Vrecica(reproved), Sarkaria (reproved), Roundeff (reproved)
Gil Kalai Discrete geometry and Jirka Matousek
Tverberg’s theorem
Tverberg’s theorem: Every set of points x1, x2, . . . , xm form = (d + 1)(r − 1) + 1 can be divided into m pairwise disjointparts X1,X2, . . . ,Xr such that
conv(X1) ∩ conv(X2),∩ · · · ∩ conv(Xr ) 6= ∅.
History: Birch (conjectured), Rado (proved a weaker result),Tverberg (proved), Tverberg (reproved), Tverberg & Vrecica(reproved), Sarkaria (reproved), Roundeff (reproved)
Gil Kalai Discrete geometry and Jirka Matousek
Topological Tverberg’s theorems and conjectures
Topological Tverberg conjecture: Let f : ∆(d+1)(r−1) → Rd bea continuous function from the (d + 1)(r − 1) dimensional simplexto Rd . Then there are r disjoint faces of the simplex whose imageshave a point in common.
History: Barany and Bajmoczy , Barany, Shlosman and Szucs...Correct for r prime. Ozaydin (and others) Correct for r primepower.Zivaljevic and Vrecica, Blagojecic, Matschke, and Ziegler
Recent breakthrough: NO! Florian Frick relying on a theory ofIsaak Mabillard and Uli Wagner.
Gil Kalai Discrete geometry and Jirka Matousek
Topological Tverberg’s theorems and conjectures
Topological Tverberg conjecture: Let f : ∆(d+1)(r−1) → Rd bea continuous function from the (d + 1)(r − 1) dimensional simplexto Rd . Then there are r disjoint faces of the simplex whose imageshave a point in common.
History: Barany and Bajmoczy , Barany, Shlosman and Szucs...Correct for r prime. Ozaydin (and others) Correct for r primepower.Zivaljevic and Vrecica, Blagojecic, Matschke, and Ziegler
Recent breakthrough: NO! Florian Frick relying on a theory ofIsaak Mabillard and Uli Wagner.
Gil Kalai Discrete geometry and Jirka Matousek
Gil Kalai Discrete geometry and Jirka Matousek
Topological Tverberg - another approach
Idea: Perhaps we should study topological Tverberg theorems viarepeated applications of Z/2Z actions rather than via Z/pZactions.
Gil Kalai Discrete geometry and Jirka Matousek
Thank you very much
Gil Kalai Discrete geometry and Jirka Matousek