On neighbors in geometric permutations

On Neighbors in Geometric Permutations

Geometric Permutations and Common Transversals

Forbidden Families of Geometric Permutations in R

A geometric permutation induced by a transversal line of a finite family F of disjoint convex sets in Rd is the order in which the transversal meets the members of the family. We prove that for each natural k, each family of k permutations is realizable (as a family of geometric permutations of some F ) in Rd for d ≥ 2k − 1, but there is a family of k permutations which is non-realizable in Rd ...

Triangulations with Very Few Geometric Bistellar Neighbors

We are interested in a notion of elementary change between triangulations of a point connguration, the so-called bistellar ips, introduced by Gel'fand, Kapranov and Zelevinski. We construct sequences of triangulations of point conngurations in dimension 3 with n 2 +2n+2 vertices and only 4n?3 geometric bistellar ips (for every even integer n), and of point conngurations in dimension 4 with arbi...

Geometric 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.