Lower bounds for the number of small convex k -holes Oswin Aichholzer - PowerPoint PPT Presentation

Institute for Software Technology Graz University of Technology P 23629–N18 Lower bounds for the number of small convex k -holes Oswin Aichholzer 1 , Ruy Fabila-Monroy 2 , Thomas Hackl 1 , Clemens Huemer 3 , Alexander Pilz 1 , and Birgit Vogtenhuber 1 1 Institute for Software Technology, Graz University of Technology 2 Departamento de Matem´ aticas, Cinvestav, Mexico City, Mexico 3 Departament de Matem` atica Aplicada IV, UPC, Barcelona, Spain Thomas Hackl: 24 th Canadian Conference on Computational Geometry, August 8 th – 10 th , 2012 1

Institute for Software Technology Graz University of Technology P 23629–N18 Definition • sets S of n points in R 2 in general position • convex k -hole P : ◦ P is a convex polygon spanned by exactly k points of S and no other point of S is contained in P • ∂ CH( S ) . . . boundary of the convex hull CH( S ) of S • ld( x ) = log x log 2 . . . binary logarithm or logarithmus dualis Thomas Hackl: 24 th Canadian Conference on Computational Geometry, August 8 th – 10 th , 2012 2

Institute for Software Technology Graz University of Technology P 23629–N18 Introduction • classical existence question by Erd˝ os: ◦ What is the smallest integer h ( k ) such that any set of h ( k ) points in R 2 contains at least one convex k -hole? • Answers: ◦ k = 4 : E. Klein: h (4) = 5 ◦ k = 5 : H. Harborth: h (5) = 10 ◦ k = 6 : T. Gerken and C. Nicol´ as: h (6) = finite ◦ k = 7 : J. Horton: ∃ arbitrary large sets without convex 7 -holes Thomas Hackl: 24 th Canadian Conference on Computational Geometry, August 8 th – 10 th , 2012 3

Institute for Software Technology Graz University of Technology P 23629–N18 Introduction • generalization of Erd˝ os’ question: ◦ What is the least number h k ( n ) of convex k -holes determined by any set of n points in R 2 ? • h k ( n ) = min | S | = n { h k ( S ) } ; we consider 3 ≤ k ≤ 5 Thomas Hackl: 24 th Canadian Conference on Computational Geometry, August 8 th – 10 th , 2012 4

Institute for Software Technology Graz University of Technology P 23629–N18 Introduction • generalization of Erd˝ os’ question: ◦ What is the least number h k ( n ) of convex k -holes determined by any set of n points in R 2 ? • h k ( n ) = min | S | = n { h k ( S ) } ; we consider 3 ≤ k ≤ 5 • h 5 ( n ) ≥ n → h 5 ( n ) ≥ 3 n 2 − O (1) [Valtr] − 4 − o ( n ) • h 3 ( n ) ≥ n 2 − 37 n 8 + 23 8 [Garc´ ıa] → h 3 ( n ) ≥ n 2 − 32 n 7 + 22 − 7 • h 4 ( n ) ≥ n 2 2 − 11 n 4 − 9 4 [Garc´ ıa] → h 4 ( n ) ≥ n 2 2 − 9 n − 4 − o ( n ) Thomas Hackl: 24 th Canadian Conference on Computational Geometry, August 8 th – 10 th , 2012 4

Institute for Software Technology Graz University of Technology P 23629–N18 Introduction • generalization of Erd˝ os’ question: ◦ What is the least number h k ( n ) of convex k -holes determined by any set of n points in R 2 ? • h k ( n ) = min | S | = n { h k ( S ) } ; we consider 3 ≤ k ≤ 5 • h 5 ( n ) ≥ n → h 5 ( n ) ≥ 3 n 2 − O (1) [Valtr] − 4 − o ( n ) • h 3 ( n ) ≥ n 2 − 37 n 8 + 23 8 [Garc´ ıa] → h 3 ( n ) ≥ n 2 − 32 n 7 + 22 − 7 • h 4 ( n ) ≥ n 2 2 − 11 n 4 − 9 4 [Garc´ ıa] → h 4 ( n ) ≥ n 2 2 − 9 n − 4 − o ( n ) Thomas Hackl: 24 th Canadian Conference on Computational Geometry, August 8 th – 10 th , 2012 4

Institute for Software Technology Graz University of Technology P 23629–N18 Convex 5-holes any and Valtr, 2004: h 5 ( n ) ≤ 1 . 0207 n 2 + o ( n 2 ) • B´ ar´ • Valtr, 2012: h 5 ( n ) ≥ n h 5 ( n ) ≥ 3 2 − O (1) 4 n − o ( n ) • for small n : ≤ 9 10 11 12 13 14 15 16 17 n h 5 ( n ) 0 1 2 3 3..4 3..6 3..9 ≥ 3 ≥ 3 Harborth, 1978 ≥ 3 ≤ 3 Dehnhardt, 1987 Aichholzer, H., and Vogtenhuber, 2012 Thomas Hackl: 24 th Canadian Conference on Computational Geometry, August 8 th – 10 th , 2012 5

Institute for Software Technology Graz University of Technology P 23629–N18 Convex 5-holes any and Valtr, 2004: h 5 ( n ) ≤ 1 . 0207 n 2 + o ( n 2 ) • B´ ar´ • Valtr, 2012: h 5 ( n ) ≥ n h 5 ( n ) ≥ 3 2 − O (1) 4 n − o ( n ) • for small n : ≤ 9 10 11 12 13 14 15 16 17 n h 5 ( n ) 0 1 2 3 3..4 3..6 3..9 ≥ 3 ≥ 3 Harborth, 1978 ≥ 4 ≥ 3 ≤ 3 Dehnhardt, 1987 Aichholzer, H., and Vogtenhuber, 2012 Thomas Hackl: 24 th Canadian Conference on Computational Geometry, August 8 th – 10 th , 2012 5

Institute for Software Technology Graz University of Technology P 23629–N18 h 5 ( n ) : Improvement for small n Let m ≥ 0 be a natural number and t ∈ { 1 , 2 , 3 } : Every set S of n = 7 · m + 9 + t points in the plane in general position contains at least h 5 ( n ) ≥ 3 m + t = 3 n − 27+4 t convex 5-holes. 7 Thomas Hackl: 24 th Canadian Conference on Computational Geometry, August 8 th – 10 th , 2012 6

Institute for Software Technology Graz University of Technology P 23629–N18 h 5 ( n ) : Improvement for small n Let m ≥ 0 be a natural number and t ∈ { 1 , 2 , 3 } : Every set S of n = 7 · m + 9 + t points in the plane in general position contains at least h 5 ( n ) ≥ 3 m + t = 3 n − 27+4 t convex 5-holes. 7 Base case, m =0 : h 5 (10) = 1 , h 5 (11) = 2 , and h 5 (12) = 3 . Thomas Hackl: 24 th Canadian Conference on Computational Geometry, August 8 th – 10 th , 2012 6

Institute for Software Technology Graz University of Technology P 23629–N18 h 5 ( n ) : Improvement for small n Let m ≥ 0 be a natural number and t ∈ { 1 , 2 , 3 } : Every set S of n = 7 · m + 9 + t points in the plane in general position contains at least h 5 ( n ) ≥ 3 m + t = 3 n − 27+4 t convex 5-holes. 7 Base case, m =0 : h 5 (10) = 1 , h 5 (11) = 2 , and h 5 (12) = 3 . Case 1/2: ∃ p ∈ ( S ∩ ∂ CH( S )) , p vertex of a convex 5 -hole h 5 ( S ) ≥ 1+ h 5 ( S \{ p } ) ≥ 1+ h 5 ( n − 1) p Thomas Hackl: 24 th Canadian Conference on Computational Geometry, August 8 th – 10 th , 2012 6

Institute for Software Technology Graz University of Technology P 23629–N18 h 5 ( n ) : Improvement for small n Let m ≥ 0 be a natural number and t ∈ { 1 , 2 , 3 } : Every set S of n = 7 · m + 9 + t points in the plane in general position contains at least h 5 ( n ) ≥ 3 m + t = 3 n − 27+4 t convex 5-holes. 7 Base case, m =0 : h 5 (10) = 1 , h 5 (11) = 2 , and h 5 (12) = 3 . Case 2/2: ∀ p ∈ ( S ∩ ∂ CH( S )) : p is not a vertex of a convex 5 -hole Thomas Hackl: 24 th Canadian Conference on Computational Geometry, August 8 th – 10 th , 2012 7

Institute for Software Technology Graz University of Technology P 23629–N18 h 5 ( n ) : Improvement for small n Let m ≥ 0 be a natural number and t ∈ { 1 , 2 , 3 } : Every set S of n = 7 · m + 9 + t points in the plane in general position contains at least h 5 ( n ) ≥ 3 m + t = 3 n − 27+4 t convex 5-holes. 7 Base case, m =0 : h 5 (10) = 1 , h 5 (11) = 2 , and h 5 (12) = 3 . Case 2/2: ∀ p ∈ ( S ∩ ∂ CH( S )) : p is not a vertex of a convex 5 -hole ( m − 1) pairs � �� � | S ′ S i S ′ 3 4 3 4 . . . . . . 0 | = 4 i | S 0 | = 7 | S rem | = t +4 p Thomas Hackl: 24 th Canadian Conference on Computational Geometry, August 8 th – 10 th , 2012 7

Institute for Software Technology Graz University of Technology P 23629–N18 h 5 ( n ) : Improvement for small n Let m ≥ 0 be a natural number and t ∈ { 1 , 2 , 3 } : Every set S of n = 7 · m + 9 + t points in the plane in general position contains at least h 5 ( n ) ≥ 3 m + t = 3 n − 27+4 t convex 5-holes. 7 Base case, m =0 : h 5 (10) = 1 , h 5 (11) = 2 , and h 5 (12) = 3 . Case 2/2: ∀ p ∈ ( S ∩ ∂ CH( S )) : p is not a vertex of a convex 5 -hole n = 1 + 7+4 + 7( m − 1) + t +4 ( m − 1) pairs � �� � | S ′ S i S ′ h 5 ( S ) = ? 3 4 3 4 . . . . . . 0 | = 4 i | S 0 | = 7 | S rem | = t +4 p Thomas Hackl: 24 th Canadian Conference on Computational Geometry, August 8 th – 10 th , 2012 7

Institute for Software Technology Graz University of Technology P 23629–N18 h 5 ( n ) : Improvement for small n Let m ≥ 0 be a natural number and t ∈ { 1 , 2 , 3 } : Every set S of n = 7 · m + 9 + t points in the plane in general position contains at least h 5 ( n ) ≥ 3 m + t = 3 n − 27+4 t convex 5-holes. 7 Base case, m =0 : h 5 (10) = 1 , h 5 (11) = 2 , and h 5 (12) = 3 . Case 2/2: ∀ p ∈ ( S ∩ ∂ CH( S )) : p is not a vertex of a convex 5 -hole n = 1 + 7+4 + 7( m − 1) + t +4 ( m − 1) pairs � �� � | S ′ S i S ′ h 5 ( S ) = ? 3 4 3 4 . . . . . . 0 | = 4 i ≥ 3 | S 0 | = 7 | S rem | = t +4 p Thomas Hackl: 24 th Canadian Conference on Computational Geometry, August 8 th – 10 th , 2012 7

Institute for Software Technology Graz University of Technology P 23629–N18 h 5 ( n ) : Improvement for small n Let m ≥ 0 be a natural number and t ∈ { 1 , 2 , 3 } : Every set S of n = 7 · m + 9 + t points in the plane in general position contains at least h 5 ( n ) ≥ 3 m + t = 3 n − 27+4 t convex 5-holes. 7 Base case, m =0 : h 5 (10) = 1 , h 5 (11) = 2 , and h 5 (12) = 3 . Case 2/2: ∀ p ∈ ( S ∩ ∂ CH( S )) : p is not a vertex of a convex 5 -hole n = 1 + 7+4 + 7( m − 1) + t +4 ( m − 1) pairs � �� � | S ′ S i S ′ h 5 ( S ) = ? 3 4 3 4 . . . . . . 0 | = 4 i ≥ 3 + 3( m − 1) | S 0 | = 7 | S rem | = t +4 p Thomas Hackl: 24 th Canadian Conference on Computational Geometry, August 8 th – 10 th , 2012 7