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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

An obstacle representation of a graph G consists of a set of pairwise disjoint simply-connected closed regions and a one-to-one mapping of the vertices of G to points such that two vertices are adjacent in G if and only if the line segment connecting the two corresponding points does not intersect any obstacle. The obstacle number of a graph is the smallest number of obstacles in an obstacle representation of the graph in the plane such that all obstacles are simple polygons.
It is known that the obstacle number of each n-vertex graph is O(n log n) [Balko, Cibulka, and Valtr, 2018] and that there are n-vertex graphs whose obstacle number is Ω(n/(log log n)²) [Dujmović and Morin, 2015]. We improve this lower bound to Ω(n/log log n) for simple polygons and to Ω(n) for convex polygons. To obtain these stronger bounds, we improve known estimates on the number of n-vertex graphs with bounded obstacle number, solving a conjecture by Dujmović and Morin. We also show that if the drawing of some n-vertex graph is given as part of the input, then for some drawings Ω(n²) obstacles are required to turn them into an obstacle representation of the graph. Our bounds are asymptotically tight in several instances.
We complement these combinatorial bounds by two complexity results. First, we show that computing the obstacle number of a graph G is fixed-parameter tractable in the vertex cover number of G. Second, we show that, given a graph G and a simple polygon P, it is NP-hard to decide whether G admits an obstacle representation using P as the only obstacle.

Martin Balko, Steven Chaplick, Robert Ganian, Siddharth Gupta, Michael Hoffmann, Pavel Valtr, and Alexander Wolff. Bounding and Computing Obstacle Numbers of Graphs. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 11:1-11:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{balko_et_al:LIPIcs.ESA.2022.11, author = {Balko, Martin and Chaplick, Steven and Ganian, Robert and Gupta, Siddharth and Hoffmann, Michael and Valtr, Pavel and Wolff, Alexander}, title = {{Bounding and Computing Obstacle Numbers of Graphs}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {11:1--11:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.11}, URN = {urn:nbn:de:0030-drops-169495}, doi = {10.4230/LIPIcs.ESA.2022.11}, annote = {Keywords: Obstacle representation, Obstacle number, Visibility, NP-hardness, FPT} }

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**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

We consider point sets in the real projective plane ℝ𝒫² and explore variants of classical extremal problems about planar point sets in this setting, with a main focus on Erdős-Szekeres-type problems.
We provide asymptotically tight bounds for a variant of the Erdős-Szekeres theorem about point sets in convex position in ℝ𝒫², which was initiated by Harborth and Möller in 1994. The notion of convex position in ℝ𝒫² agrees with the definition of convex sets introduced by Steinitz in 1913.
For k ≥ 3, an (affine) k-hole in a finite set S ⊆ ℝ² is a set of k points from S in convex position with no point of S in the interior of their convex hull. After introducing a new notion of k-holes for points sets from ℝ𝒫², called projective k-holes, we find arbitrarily large finite sets of points from ℝ𝒫² with no projective 8-holes, providing an analogue of a classical result by Horton from 1983. We also prove that they contain only quadratically many projective k-holes for k ≤ 7. On the other hand, we show that the number of k-holes can be substantially larger in ℝ𝒫² than in ℝ² by constructing, for every k ∈ {3,… ,6}, sets of n points from ℝ² ⊂ ℝ𝒫² with Ω(n^{3-3/5k}) projective k-holes and only O(n²) affine k-holes. Last but not least, we prove several other results, for example about projective holes in random point sets in ℝ𝒫² and about some algorithmic aspects.
The study of extremal problems about point sets in ℝ𝒫² opens a new area of research, which we support by posing several open problems.

Martin Balko, Manfred Scheucher, and Pavel Valtr. Erdős-Szekeres-Type Problems in the Real Projective Plane. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{balko_et_al:LIPIcs.SoCG.2022.10, author = {Balko, Martin and Scheucher, Manfred and Valtr, Pavel}, title = {{Erd\H{o}s-Szekeres-Type Problems in the Real Projective Plane}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {10:1--10:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.10}, URN = {urn:nbn:de:0030-drops-160182}, doi = {10.4230/LIPIcs.SoCG.2022.10}, annote = {Keywords: real projective plane, point set, convex position, k-gon, k-hole, Erd\H{o}s-Szekeres theorem, Horton set, random point set} }

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**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

For a finite set A ⊂ ℝ², a map φ: A → ℝ² is orientation preserving if for every non-collinear triple u,v,w ∈ A the orientation of the triangle u,v,w is the same as that of the triangle φ(u),φ(v),φ(w). We prove that for every n ∈ ℕ and for every ε > 0 there is N = N(n,ε) ∈ ℕ such that the following holds. Assume that φ:G(N) → ℝ² is an orientation preserving map where G(N) is the grid {(i,j) ∈ ℤ²: -N ≤ i,j ≤ N}. Then there is an affine transformation ψ :ℝ² → ℝ² and a ∈ ℤ² such that a+G(n) ⊂ G(N) and ‖ψ∘φ (z)-z‖ < ε for every z ∈ a+G(n). This result was previously proved in a completely different way by Nešetřil and Valtr, without obtaining any bound on N. Our proof gives N(n,ε) = O(n⁴ε^{-2}).

Imre Bárány, Attila Pór, and Pavel Valtr. Orientation Preserving Maps of the Square Grid. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 14:1-14:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{barany_et_al:LIPIcs.SoCG.2021.14, author = {B\'{a}r\'{a}ny, Imre and P\'{o}r, Attila and Valtr, Pavel}, title = {{Orientation Preserving Maps of the Square Grid}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {14:1--14:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.14}, URN = {urn:nbn:de:0030-drops-138130}, doi = {10.4230/LIPIcs.SoCG.2021.14}, annote = {Keywords: square grid, plane, order type} }

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**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

For d ∈ ℕ, let S be a finite set of points in ℝ^d in general position. A set H of k points from S is a k-hole in S if all points from H lie on the boundary of the convex hull conv(H) of H and the interior of conv(H) does not contain any point from S. A set I of k points from S is a k-island in S if conv(I) ∩ S = I. Note that each k-hole in S is a k-island in S.
For fixed positive integers d, k and a convex body K in ℝ^d with d-dimensional Lebesgue measure 1, let S be a set of n points chosen uniformly and independently at random from K. We show that the expected number of k-islands in S is in O(n^d). In the case k=d+1, we prove that the expected number of empty simplices (that is, (d+1)-holes) in S is at most 2^(d-1) ⋅ d! ⋅ binom(n,d). Our results improve and generalize previous bounds by Bárány and Füredi [I. Bárány and Z. Füredi, 1987], Valtr [P. Valtr, 1995], Fabila-Monroy and Huemer [Fabila-Monroy and Huemer, 2012], and Fabila-Monroy, Huemer, and Mitsche [Fabila-Monroy et al., 2015].

Martin Balko, Manfred Scheucher, and Pavel Valtr. Holes and Islands in Random Point Sets. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 14:1-14:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{balko_et_al:LIPIcs.SoCG.2020.14, author = {Balko, Martin and Scheucher, Manfred and Valtr, Pavel}, title = {{Holes and Islands in Random Point Sets}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {14:1--14:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.14}, URN = {urn:nbn:de:0030-drops-121722}, doi = {10.4230/LIPIcs.SoCG.2020.14}, annote = {Keywords: stochastic geometry, random point set, Erd\H{o}s-Szekeres type problem, k-hole, k-island, empty polytope, convex position, Horton set} }

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**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Let P be a set of 2n points in convex position, such that n points are colored red and n points are colored blue. A non-crossing alternating path on P of length 𝓁 is a sequence p₁, … , p_𝓁 of 𝓁 points from P so that (i) all points are pairwise distinct; (ii) any two consecutive points p_i, p_{i+1} have different colors; and (iii) any two segments p_i p_{i+1} and p_j p_{j+1} have disjoint relative interiors, for i ≠ j.
We show that there is an absolute constant ε > 0, independent of n and of the coloring, such that P always admits a non-crossing alternating path of length at least (1 + ε)n. The result is obtained through a slightly stronger statement: there always exists a non-crossing bichromatic separated matching on at least (1 + ε)n points of P. This is a properly colored matching whose segments are pairwise disjoint and intersected by common line. For both versions, this is the first improvement of the easily obtained lower bound of n by an additive term linear in n. The best known published upper bounds are asymptotically of order 4n/3+o(n).

Wolfgang Mulzer and Pavel Valtr. Long Alternating Paths Exist. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 57:1-57:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{mulzer_et_al:LIPIcs.SoCG.2020.57, author = {Mulzer, Wolfgang and Valtr, Pavel}, title = {{Long Alternating Paths Exist}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {57:1--57:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.57}, URN = {urn:nbn:de:0030-drops-122152}, doi = {10.4230/LIPIcs.SoCG.2020.57}, annote = {Keywords: Non-crossing path, bichromatic point sets} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

The Tverberg theorem is one of the cornerstones of discrete geometry. It states that, given a set X of at least (d+1)(r-1)+1 points in R^d, one can find a partition X=X_1 cup ... cup X_r of X, such that the convex hulls of the X_i, i=1,...,r, all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed.
As a concrete application, we show that any n points in the plane in general position span floor[n/3] vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Alvarez-Rebollar et al. guarantees floor[n/6] pairwise crossing triangles. Our result generalizes to a result about simplices in R^d,d >=2.

Radoslav Fulek, Bernd Gärtner, Andrey Kupavskii, Pavel Valtr, and Uli Wagner. The Crossing Tverberg Theorem. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 38:1-38:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fulek_et_al:LIPIcs.SoCG.2019.38, author = {Fulek, Radoslav and G\"{a}rtner, Bernd and Kupavskii, Andrey and Valtr, Pavel and Wagner, Uli}, title = {{The Crossing Tverberg Theorem}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {38:1--38:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.38}, URN = {urn:nbn:de:0030-drops-104423}, doi = {10.4230/LIPIcs.SoCG.2019.38}, annote = {Keywords: Discrete geometry, Tverberg theorem, Crossing Tverberg theorem} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

A permutation pi is a merge of a permutation sigma and a permutation tau, if we can color the elements of pi red and blue so that the red elements have the same relative order as sigma and the blue ones as tau. We consider, for fixed hereditary permutation classes C and D, the complexity of determining whether a given permutation pi is a merge of an element of C with an element of D.
We develop general algorithmic approaches for identifying polynomially tractable cases of merge recognition. Our tools include a version of nondeterministic logspace streaming recognizability of permutations, which we introduce, and a concept of bounded width decomposition, inspired by the work of Ahal and Rabinovich.
As a consequence of the general results, we can provide nontrivial examples of tractable permutation merges involving commonly studied permutation classes, such as the class of layered permutations, the class of separable permutations, or the class of permutations avoiding a decreasing sequence of a given length.
On the negative side, we obtain a general hardness result which implies, for example, that it is NP-complete to recognize the permutations that can be merged from two subpermutations avoiding the pattern 2413.

Vít Jelínek, Michal Opler, and Pavel Valtr. Generalized Coloring of Permutations. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 50:1-50:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{jelinek_et_al:LIPIcs.ESA.2018.50, author = {Jel{\'\i}nek, V{\'\i}t and Opler, Michal and Valtr, Pavel}, title = {{Generalized Coloring of Permutations}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {50:1--50:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.50}, URN = {urn:nbn:de:0030-drops-95137}, doi = {10.4230/LIPIcs.ESA.2018.50}, annote = {Keywords: Permutations, merge, generalized coloring} }

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**Published in:** LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Let P be a finite set of points in the plane in general position, that is, no three points of P are on a common line. We say that a set H of five points from P is a 5-hole in P if H is the vertex set of a convex 5-gon containing no other points of P. For a positive integer n, let h_5(n) be the minimum number of 5-holes among all sets of n points in the plane in general position.
Despite many efforts in the last 30 years, the best known asymptotic lower and upper bounds for h_5(n) have been of order Omega(n) and O(n^2), respectively. We show that h_5(n) = Omega(n(log n)^(4/5)), obtaining the first superlinear lower bound on h_5(n).
The following structural result, which might be of independent interest, is a crucial step in the proof of this lower bound. If a finite set P of points in the plane in general position is partitioned by a line l into two subsets, each of size at least 5 and not in convex position, then l intersects the convex hull of some 5-hole in P. The proof of this result is computer-assisted.

Oswin Aichholzer, Martin Balko, Thomas Hackl, Jan Kyncl, Irene Parada, Manfred Scheucher, Pavel Valtr, and Birgit Vogtenhuber. A Superlinear Lower Bound on the Number of 5-Holes. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{aichholzer_et_al:LIPIcs.SoCG.2017.8, author = {Aichholzer, Oswin and Balko, Martin and Hackl, Thomas and Kyncl, Jan and Parada, Irene and Scheucher, Manfred and Valtr, Pavel and Vogtenhuber, Birgit}, title = {{A Superlinear Lower Bound on the Number of 5-Holes}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {8:1--8:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.8}, URN = {urn:nbn:de:0030-drops-72008}, doi = {10.4230/LIPIcs.SoCG.2017.8}, annote = {Keywords: Erd\"{o}s-Szekeres type problem, k-hole, empty k-gon, empty pentagon, planar point set} }

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**Published in:** LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Let d and k be integers with 1 <= k <= d-1. Let Lambda be a d-dimensional lattice and let K be a d-dimensional compact convex body symmetric about the origin. We provide estimates for the minimum number of k-dimensional linear subspaces needed to cover all points in the intersection of Lambda with K. In particular, our results imply that the minimum number of k-dimensional linear subspaces needed to cover the d-dimensional n * ... * n grid is at least Omega(n^(d(d-k)/(d-1)-epsilon)) and at most O(n^(d(d-k)/(d-1))), where epsilon > 0 is an arbitrarily small constant. This nearly settles a problem mentioned in the book of Brass, Moser, and Pach. We also find tight bounds for the minimum number of k-dimensional affine subspaces needed to cover the intersection of Lambda with K.
We use these new results to improve the best known lower bound for the maximum number of point-hyperplane incidences by Brass and Knauer. For d > =3 and epsilon in (0,1), we show that there is an integer r=r(d,epsilon) such that for all positive integers n, m the following statement is true. There is a set of n points in R^d and an arrangement of m hyperplanes in R^d with no K_(r,r) in their incidence graph and with at least Omega((mn)^(1-(2d+3)/((d+2)(d+3)) - epsilon)) incidences if d is odd and Omega((mn)^(1-(2d^2+d-2)/((d+2)(d^2+2d-2)) - epsilon)) incidences if d is even.

Martin Balko, Josef Cibulka, and Pavel Valtr. Covering Lattice Points by Subspaces and Counting Point-Hyperplane Incidences. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{balko_et_al:LIPIcs.SoCG.2017.12, author = {Balko, Martin and Cibulka, Josef and Valtr, Pavel}, title = {{Covering Lattice Points by Subspaces and Counting Point-Hyperplane Incidences}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {12:1--12:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.12}, URN = {urn:nbn:de:0030-drops-71955}, doi = {10.4230/LIPIcs.SoCG.2017.12}, annote = {Keywords: lattice point, covering, linear subspace, point-hyperplane incidence} }

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**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Let S be a subset of R^d with finite positive Lebesgue measure. The Beer index of convexity b(S) of S is the probability that two points of S chosen uniformly independently at random see each other in S. The convexity ratio c(S) of S is the Lebesgue measure of the largest convex subset of S divided by the Lebesgue measure of S. We investigate a relationship between these two natural measures of convexity of S.
We show that every subset S of the plane with simply connected components satisfies b(S) <= alpha c(S) for an absolute constant alpha, provided b(S) is defined. This implies an affirmative answer to the conjecture of Cabello et al. asserting that this estimate holds for simple polygons.
We also consider higher-order generalizations of b(S). For 1 <= k <= d, the k-index of convexity b_k(S) of a subset S of R^d is the probability that the convex hull of a (k+1)-tuple of points chosen uniformly independently at random from S is contained in S. We show that for every d >= 2 there is a constant beta(d) > 0 such that every subset S of R^d satisfies b_d(S) <= beta c(S), provided b_d(S) exists. We provide an almost matching lower bound by showing that there is a constant gamma(d) > 0 such that for every epsilon from (0,1] there is a subset S of R^d of Lebesgue measure one satisfying c(S) <= epsilon and b_d(S) >= (gamma epsilon)/log_2(1/epsilon) >= (gamma c(S))/log_2(1/c(S)).

Martin Balko, Vít Jelínek, Pavel Valtr, and Bartosz Walczak. On the Beer Index of Convexity and Its Variants. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 406-420, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{balko_et_al:LIPIcs.SOCG.2015.406, author = {Balko, Martin and Jel{\'\i}nek, V{\'\i}t and Valtr, Pavel and Walczak, Bartosz}, title = {{On the Beer Index of Convexity and Its Variants}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {406--420}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.406}, URN = {urn:nbn:de:0030-drops-51229}, doi = {10.4230/LIPIcs.SOCG.2015.406}, annote = {Keywords: Beer index of convexity, convexity ratio, convexity measure, visibility} }