Helly-Type Theorems for Line Transversals to Disjoint Unit Balls
نویسندگان
چکیده
We prove Helly-type theorems for line transversals to disjoint unit balls in R. In particular, we show that a family of n > 2d disjoint unit balls in R has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with ≺. We also prove that a family of n > 4d − 1 disjoint unit balls in R admits a line transversal if any subfamily of size 4d − 1 admits a transversal.
منابع مشابه
Hadwiger and Helly-type theorems for disjoint unit spheres
We prove Helly-type theorems for line transversals to disjoint unit balls in R. In particular, we show that a family of n > 2d disjoint unit balls in R has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with ≺. We also prove that a family of n > 4d − 1 disjoint unit balls in R admits a line transversal if any subfamily of ...
متن کاملOn the Helly Number for Hyperplane Transversals to Unit Balls
We prove some results about the Hadwiger problem of nding the Helly number for line transversals of disjoint unit disks in the plane, and about its higher-dimensional generalization to hyperplane transversals of unit balls in d-dimensional Euclidean space. These include (a) a proof of the fact that the Helly number remains 5 even for arbitrarily large sets of disjoint unit disks|thus correcting...
متن کاملSome Discrete Properties of the Space of Line Transversals to Disjoint Balls
Attempts to generalize Helly’s theorem to sets of lines intersecting convex sets led to a series of results relating the geometry of a family of sets in Rd to the structure of the space of lines intersecting all of its members. We review recent progress in the special case of disjoint Euclidean balls in Rd, more precisely the inter-related notions of cone of directions, geometric permutations a...
متن کاملThe Harmony of Spheres
Let F = {X1, . . . , Xn} be a family of disjoint compact convex sets in R. An oriented straight line ` that intersects every Xi is called a (line) transversal to F . A transversal naturally induces an ordering of the elements of F , this ordering, together with its reverse (which is induced by the same line with opposite orientation), is called a geometric permutation. Geometric transversal the...
متن کاملGeometric permutations of disjoint unit spheres
We show that a set of n disjoint unit spheres in R admits at most two distinct geometric permutations if n ≥ 9, and at most three if 3 ≤ n ≤ 8. This result improves a Helly-type theorem on line transversals for disjoint unit spheres in R: if any subset of size 18 of a family of such spheres admits a line transversal, then there is a line transversal for the entire family.
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 & Computational Geometry
دوره 29 شماره
صفحات -
تاریخ انتشار 2003