Berge's theorem, fractional Helly, and art galleries
نویسندگان
چکیده
In one of his early papers Claude Berge proved a Helly-type theorem, which replaces the usual “nonempty intersection” condition with a “convex union” condition. Inspired by this we prove a fractional Helly-type result, where we assume that many (d+1)-tuples of a family of convex sets have a star-shaped union, and the conclusion is that many of the sets have a common point. We also investigate somewhat related art-gallery problems. In particular, we prove a (p, 3)-theorem for guarding planar art galleries with a bounded number of holes, completing a result of Kalai and Matoušek, who obtained such a result for galleries without holes. On the other hand, we show that if the number of holes is unbounded, then no (p, q)-theorem of this kind holds with p 2q − 1. © 2006 Elsevier B.V. All rights reserved.
منابع مشابه
Bounded VC-Dimension Implies a Fractional Helly Theorem
We prove that every set system of bounded VC-dimension has a fractional Helly property. More precisely, if the dual shatter function of a set system F is bounded by o(m k), then F has fractional Helly number k. This means that for every > 0 there exists a > 0 such that if F 1 ; F 2 ; : : : ; F n 2 F are sets with T i2I F i 6 = ; for at least ? n k sets I f1; 2; : : :; ng of size k, then there e...
متن کاملHigher order multi-point fractional boundary value problems with integral boundary conditions
In this paper, we concerned with positive solutions for higher order m-point nonlinear fractional boundary value problems with integral boundary conditions. We establish the criteria for the existence of at least one, two and three positive solutions for higher order m-point nonlinear fractional boundary value problems with integral boundary conditions by using a result from the theory of fixed...
متن کاملHelly numbers of acyclic families
The Helly number of a family of sets with empty intersection is the size of its largest inclusionwise minimal sub-family with empty intersection. Let F be a finite family of open subsets of an arbitrary locally arc-wise connected topological space Γ. Assume that for every sub-family G ⊆ F the intersection of the elements of G has at most r connected components, each of which is a Q-homology cel...
متن کاملA fractional Helly theorem for convex lattice sets
A set of the form C-Z ; where CDR is convex and Z denotes the integer lattice, is called a convex lattice set. It is known that the Helly number of d-dimensional convex lattice sets is 2 : We prove that the fractional Helly number is only d þ 1: For every d and every aAð0; 1 there exists b40 such that whenever F1;y;Fn are convex lattice sets in Z such that T iAI Fia| for at least að n dþ1Þ inde...
متن کاملA Note on Smaller Fractional Helly Numbers
Let F be a family of geometric objects in R such that the complexity (number of faces of all dimensions on the boundary) of the union of any m of them is o(m). We show that F , as well as {F ∩P | F ∈ F} for any given set P ∈ R, have fractional Helly number at most k. This improves the known bounds for fractional Helly numbers of many families.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Discrete Mathematics
دوره 306 شماره
صفحات -
تاریخ انتشار 2006