discrete geometry - personal reflections on some works by jirí … · i phenomena in discrete...

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.

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.

1: Linear Programming

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!

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.

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√

If an+1 − an = Median(a1, a2 . . . , an) then an = exp(K log2 n).

I also discovered that nlog d = d log n.

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√

If an+1 − an = Median(a1, a2 . . . , an) then an = exp(K log2 n).

I also discovered that nlog d = d log n.

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.

Random Facet can be exponential (in√

d) for abstract cubes

This is an early result from 1994 by Jiri 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!

Random edge can be exponential(*) for abstract cubes

This is a result by Jiri Matousek and Tibor Szabo from 2004.

Recent breakthrough: Random edge and random facet canbe exponential(*) for LP

This is a 2010 breakthrough by Oliver Friedmann, Thomas Hansen,and Uri Zwick

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

Part 2: Our art gallery theorem

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.)

Part 3: Helly theorem, and the fractional Helly theorem

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.

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.

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.

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?

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.

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.

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.

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.

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.

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.

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.

The case k = 0

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).

The case k = 0

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).

Part 4: Topological methods (mainly a la Borsuk)

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)

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)

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.

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.

Topological Tverberg - another approach

Idea: Perhaps we should study topological Tverberg theorems viarepeated applications of Z/2Z actions rather than via Z/pZactions.

Thank you very much

discrete geometry - personal reflections on some works by jirí … · i phenomena in discrete...

Documents