Guest Editors' Foreword

Imre Bárány, Luis Montejano, Deborah Oliveros
2010 Discrete & Computational Geometry  
This special Issue of Discrete & Computational Geometry is dedicated to the presentation of some recent results in the area of geometric transversals and Helly-type theorems. The idea of the special issue grew out of the highly successful workshop, called Transversal and Helly-type Theorems in Geometry, Combinatorics, and Topology, which took place at the Banff International Research Station (BIRS) between the 20th and the 25th of September, 2009. This issue contains 12 articles, and all of
more » ... were refereed according to the usual high standards of Discrete & Computational Geometry. Helly's theorem is perhaps one of the most cited theorems in discrete geometry and has stimulated numerous generalizations and variants. The search for Helly numbers in different "universes" other than convex sets has inspired many articles; for instance in this volume, J.L. Arocha and J. Bracho show that the lattice of linear partitions in a projective geometry of rank n has a Helly number. There are many interesting connections between Helly's theorem and its relatives, the theorems of Radon, of Carathéodory, and of Tverberg. In fact, one of the most beautiful theorems in combinatorial convexity is Tverberg's theorem, which is the r-partite version of Radon's theorem, and it is very closely connected to the multiplied, or colorful, versions of the theorems of Helly, Hadwiger, and Carathéodory. In this spirit, J. Bokowski, J. Bracho and R. Strausz present some generalization of the Bárány-Carathéodory theorem to oriented matroids of Euclidean dimension 3. R. Strausz introduces the notion of hyperseparoids as a generalization of separoids, and proves a representation theorem for hyperseparoids in terms of Tverberg partitions. I. Bárány ( ) Rényi Institute,
doi:10.1007/s00454-010-9289-5 fatcat:ejgblstegjal7gwjj77tqjeopa