Discrete geometry - Personal reflections on some

Transcription

1 Discrete geometry - Personal reflections on some works by Jiří Matoušek June 23, 2015 LFT 100 meeting, Budapest 2015.

2 Some general themes for this lecture Graphs and hypergraphs arising in geometry are very special

3 Some general themes for this lecture Graphs and hypergraphs arising in geometry are very special Phenomena in discrete geometry often have have strong topological flavour.

4 Some general themes for this lecture Graphs and hypergraphs arising in geometry are very special Phenomena in discrete geometry often have have strong topological flavour. Phenomena in discrete geometry often have very general combinatorial underlying explanation.

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6 A new result about approximations of smooth convex bodies by polytopes It is known that if a simplicial convex polytope P ɛ-approximates a C 2 -convex body K. Then, the number of vertices of P is Ω(ɛ (d 1)/2 ). Theorem: Adiprasito, Nevo and Samper g k (P) = Ω(ɛ (d 1)/2 ). This proves a conjecture I made in the 90s.

7 1: Linear Programming

8 A randomized simplex algorihms: Random Facet RandomFacet (Sharir and Welzl) Start from a vertex v and choose a random facet containing it Apply the algorithm recursively inside this facet Repeat!

9 Ranom Facet is subexponential! Theorem (Matoušek, Sharir, Welzl and Kalai, 1992): Random Facet requires a subexponential expected running time for every LP problem with d variables and n ineqialities. Expected number of pivot steps e K log dn.

10 A crucial fact in analysis Let a 1 < a 2 < a 3... be a monotone sequence of reals: If a n+1 a n = Average(a 1, a 2..., a n ) then a n = exp(k n). If a n+1 a n = Median(a 1, a 2..., a n ) then a n = exp(k log 2 n).

11 A crucial fact in analysis Let a 1 < a 2 < a 3... be a monotone sequence of reals: If a n+1 a n = Average(a 1, a 2..., a n ) then a n = exp(k n). If a n+1 a n = Median(a 1, a 2..., a n ) then a n = exp(k log 2 n). I also discovered that n log d = d log n.

12 Abstraction of LP 1. Abstract objective function on polytopes: unique sink (local maximum) on every face 2. Polytopes can be replaced by more abstract objects as well.

13 Random Facet can be exponential (in d) for abstract cubes This is an early result from 1994 by Jiři Matoušek.

14 Random Edge Start from a vertex v and choose a random edge containing it Move to the other vertex if this improves matters Repeat!

15 Random edge can be exponential(*) for abstract cubes This is a result by Jiři Matoušek and Tibor Szabo from 2004.

16 Recent breakthrough: Random edge and random facet can be exponential(*) for LP This is a 2010 breakthrough by Oliver Friedmann, Thomas Hansen, and Uri Zwick

17 Challenges Better pivot rules: What about RandomFace algorithm? What about random-walk based algorithms? Unique sink orientations Diameter of polytopes and abstract polytopes (the polynomial Hirsch conjecture) Can Geometry help? Average (and smoothed) case of randomized pivot rules

18 Part 2: Our art gallery theorem

19 Theorem (Kalai and Matoušek 1997: For a simply connected planar gallery of area 1, if a guard in every location sees points of area ɛ then Cɛ log(1/ɛ) guards suffices. Key: Bounded VC dimension and ɛ-nets.

20 Theorem (Kalai and Matoušek 1997: For a simply connected planar gallery of area 1, if a guard in every location sees points of area ɛ then Cɛ log(1/ɛ) guards suffices. Key: Bounded VC dimension and ɛ-nets. Question: Can we get rid of log(1/ɛ)

21 Theorem (Kalai and Matoušek 1997: For a simply connected planar gallery of area 1, if a guard in every location sees points of area ɛ then Cɛ log(1/ɛ) guards suffices. Key: Bounded VC dimension and ɛ-nets. Question: Can we get rid of log(1/ɛ) Major general question: For bounded VC dimension when can we get read of the log(1/ɛ) (related to many things, e.g., to a famous conjecture by Danzer.)

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23 Part 3: Helly theorem, and the fractional Helly theorem

24 Helly numbers and Helly s theorem A family F of sets has Helly number k if for every finite subfamily G 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 R d has Helly number d + 1.

25 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 in F, 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.

26 Topological Helly s theorem Topological Helly s theorem: (proved by Helly himself!) The class of compact sets homehomorphic to a ball in R d have Helly order d + 1.

27 Helly orders for sets with bounded complexity For a compact set K in R d let b(k) be the minimal number such that K can be presented as the union of b(k) compact convex sets. Let b 0 (K) be the minimum number so that K can be presented as the union of disjoint convex sets. Theorem (Matoušek and Alon and Kalai (around 1995)) : The class of compact sets K in R d with b(k) b have bounded Helly order.

28 Helly orders for sets with bounded complexity For a compact set K in R d let b(k) be the minimal number such that K can be presented as the union of b(k) compact convex sets. Let b 0 (K) be the minimum number so that K can be presented as the union of disjoint convex sets. Theorem (Matoušek and Alon and Kalai (around 1995)) : The class of compact sets K in R d with b(k) b have bounded Helly order. Theorem (Amenta (following Motzkin-Grunbaum, Larman and Morris)) : The class of compact sets K in R d with b 0 (K) b have Helly order b(d + 1).

29 Helly orders for sets with bounded complexity For a compact set K in R d let b(k) be the minimal number such that K can be presented as the union of b(k) compact convex sets. Let b 0 (K) be the minimum number so that K can be presented as the union of disjoint convex sets. Theorem (Matoušek and Alon and Kalai (around 1995)) : The class of compact sets K in R d with b(k) b have bounded Helly order. Theorem (Amenta (following Motzkin-Grunbaum, Larman and Morris)) : The class of compact sets K in R d with b 0 (K) b have Helly order b(d + 1). Curious question: If n > d + 1 and X 1,... X n compact sets in R d such that every j, j < n the intersection of every j sets is the union of two closed nonempty convex sets, is there always a point in common to all sets?

30 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 every subfamily G of n sets if a fraction α of all k-subfamilies are intersecting then a fraction β of all members of G have nonempty intersection. The strong FHP with index k: Also α 1 when β 1.

31 The piercing property Piercing property with index k: For every p > k there is f (p) such that if from every p sets, k sets have a point in common then there are f (p) points such that every set contains one of them.

32 Theorem (Katchalski and Liu, Eckhoff, Kalai) around 1980: Convex sets in R d have the strong fractional Helly property with index d + 1. Theorem (Alon and Kleitman, 1992): Convex sets in R d have the piercing property with index d + 1. Theorem (Alon, Kalai, Matoušek, Meshulam, 2002): The fractional Helly property implies piercing property with the same parameter.

33 The Bárány-Matoušek theorem Integral Helly theorem (Scarf): Let F be a collection of n convex sets in R d. If every 2 d sets in F have an integer point in common then there is an integer point common to all of the sets. In other words: integral convex sets in R d have Helly number 2 d.

34 The Bárány-Matoušek theorem Integral Helly theorem (Scarf): Let F be a collection of n convex sets in R d. If every 2 d sets in F have an integer point in common then there is an integer point common to all of the sets. In other words: integral convex sets in R d have Helly number 2 d. Bárány-Matoušek fractional Helly Theorem: Integral convex sets in R d satisfy the fractional Helly property with parameter d+1. In particular: There is a positive constant α(d) such that the following statement holds: Let F be a collection of n convex sets in R d. If every d + 1 sets in F have an integer point in common, then there is an integer point common to α(d)n of the sets.

35 What type of properties implies fractional Helly? Theorem: (Matoušek) Bounded VC-dimension implies the fractional Helly property. This inspired the following: Conjecture (Kalai and Meshulam): Fractional Helly of parameter k follows from polynomial growth (like n k ) of the total Betti numbers of the nerve.

36 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 induced subgraph then χ(g) is bounded.

37 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 induced subgraph then χ(g) is bounded. What about K=1. Conjecture: β(i (H)) 1 for every induced subgraph H iff G does not contain an induced cycle of length 0(mod 3).

38 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 induced subgraph then χ(g) is bounded. What about K=1. Conjecture: β(i (H)) 1 for every induced subgraph H iff G does not contain an induced cycle of length 0(mod 3). Gyárfás type question (Kalai and Meshulam): Is there a uniform upper bound for the chromatic number of all graphs G such that all induced cycles in G are of length 1 or 2 modulo 3? Answer: Yes! Theorem by Bonamy, Charbitz and Thomassé.

39 Part 4: Topological methods (mainly a la Borsuk)

40 Tverberg s theorem Tverberg s theorem: Every set of points x 1, x 2,..., x m for m = (d + 1)(r 1) + 1 can be divided into m pairwise disjoint parts X 1, X 2,..., X r such that conv(x 1 ) conv(x 2 ), conv(x r ).

41 Tverberg s theorem Tverberg s theorem: Every set of points x 1, x 2,..., x m for m = (d + 1)(r 1) + 1 can be divided into m pairwise disjoint parts X 1, X 2,..., X r such that conv(x 1 ) conv(x 2 ), conv(x r ). History: Birch (conjectured), Rado (proved a weaker result), Tverberg (proved), Tverberg (reproved), Tverberg & Vrecica (reproved), Sarkaria (reproved), Roundeff (reproved)

42 Topological Tverberg s theorems and conjectures Topological Tverberg conjecture: Let f : (d+1)(r 1) R d be a continuous function from the (d + 1)(r 1) dimensional simplex to R d. Then there are r disjoint faces of the simplex whose images have a point in common. History: Bárány and Bajmóczy, Bárány, Shlosman and Szücs... Correct for r prime. Özaydin (and others) Correct for r prime power. Zivaljevic and Vrecica, Blagojecic, Matschke, and Ziegler

43 Topological Tverberg s theorems and conjectures Topological Tverberg conjecture: Let f : (d+1)(r 1) R d be a continuous function from the (d + 1)(r 1) dimensional simplex to R d. Then there are r disjoint faces of the simplex whose images have a point in common. History: Bárány and Bajmóczy, Bárány, Shlosman and Szücs... Correct for r prime. Özaydin (and others) Correct for r prime power. Zivaljevic and Vrecica, Blagojecic, Matschke, and Ziegler Recent breakthrough: NO! Florian Frick relying on a theory of Isaak Mabillard and Uli Wagner.

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45 Topological Tverberg - another approach Idea: Perhaps we should study topological Tverberg theorems via repeated applications of Z/2Z actions rather than via Z/pZ actions.

46 Thank you very much