Hadwiger and Helly-Type Theorems for Disjoint Unit Spheres in R
نویسندگان
چکیده
Let S be an ordered set of disjoint unit spheres in R. We show that if every subset of at most six spheres from S admits a line transversal respecting the ordering, then the entire family has a line transversal. Without the order condition, we show that the existence of a line transversal for every subset of at most 11 spheres from S implies the existence of a line transversal for S.
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 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 کاملHelly-Type Theorems for Line Transversals to Disjoint Unit Balls
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 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 کامل