Disjoint Common Transversals and Exchange Structures

Disjoint Common Partial Transversals of Two Families of Sets

On 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...

Helly-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 ...

On the Connected Components of the Space of Line Transersals t a Family of Convex Sets

Let L be the space of line transversals to a nite family of pairwise disjoint compact convex sets in R 3. We prove that each connected component of L can itself be represented as the space of transversals to some nite family of pairwise disjoint compact convex sets.

The 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...