@inproceedings{32bd81dd540b4c88a9c57a133a085bba,

title = "On the Beer Index of Convexity and Its Variants",

abstract = "Let S be a subset of ℝd 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 the relationship between these two natural measures of convexity of S. We show that every set S ⊆ ℝ2 with simply connected components satisfies b(S) ≤ α c(S) for an absolute constant α, 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 ≤ κ ≤ d, the k-index of convexity bk(S) of S ⊆ ℝd is the probability that the convex hull of a (κ+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 β(d)>0 such that every set S ⊆ ℝd satisfies bd(S) ≤ βc(S), provided bd(S) exists. We provide an almost matching lower bound by showing that there is a constant γ(d)>0 such that for every ε ε (0, 1] there is a set S ⊆ ℝd of Lebesgue measure one satisfying c(S) ≤ ⊆ and (eqution presented).",

keywords = "Beer index of convexity, Convexity measure, Convexity ratio, Visibility",

author = "Martin Balko and V{\'i}t Jel{\'i}nek and Pavel Valtr and Bartosz Walczak",

year = "2015",

month = jun,

day = "1",

doi = "10.4230/LIPIcs.SOCG.2015.406",

language = "English",

series = "Leibniz International Proceedings in Informatics, LIPIcs",

publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",

pages = "406--420",

editor = "Janos Pach and Janos Pach and Lars Arge",

booktitle = "31st International Symposium on Computational Geometry, SoCG 2015",

address = "Germany",

note = "31st International Symposium on Computational Geometry, SoCG 2015 ; Conference date: 22-06-2015 Through 25-06-2015",

}