Let S be an ordered set of disjoint unit spheres in R3 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 lineâ€¦ (More)

INTRODUCTION Let F be a family of convex sets in R. A geometric transversal is an affine subspace that intersects every member of F . More specifically, for a given integer 0 â‰¤ k < d, a k-dimensionalâ€¦ (More)

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 â‰ºâ€¦ (More)

A line meeting a family of pairwise disjoint convex sets induces two permutations of the sets. This pair of permutations is called a geometric permutation. We characterize the possible triples ofâ€¦ (More)

A line l is a transversal to a family F of convex objects in R if it intersects every member of F . In this paper we show that for every integer d â‰¥ 3 there exists a family of 2d âˆ’ 1 pairwiseâ€¦ (More)

In 1940, Luis SantalÃ³ proved a Helly-type theorem for line transversals to boxes in R. An analysis of his proof reveals a convexity structure for ascending lines in R that is isomorphic to theâ€¦ (More)