Skip Navigation


Bulletin of the London Mathematical Society Advance Access originally published online on June 14, 2009
Bulletin of the London Mathematical Society 2009 41(4):577-588; doi:10.1112/blms/bdp024
This Article
Right arrow FREE Full Text (PDF) Freely available
Right arrow All Versions of this Article:
41/4/577    most recent
bdp024v1
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Eckhoff, J.
Right arrow Articles by Nischke, K.-P.
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© 2009 London Mathematical Society

Morris's pigeonhole principle and the Helly theorem for unions of convex sets

Jürgen Eckhoff

Fachbereich Mathematik
Universität Dortmund
D-44221 Dortmund
Germany

Klaus-Peter Nischke

Pfahlweg 2
D-65307 Bad Schwalbach-Lindschied
Germany
klaus-peter.nischke@uni-dortmund.de

Received 1 March 2008. Revision received 16 December 2008.

In 1973, H. C. Morris devised a combinatorial scheme, a ‘generalized pigeonhole principle’, which he used to prove a conjecture of Grünbaum and Motzkin from 1961. This conjecture proposed, in an abstract setting, a Helly-type theorem for certain families of disjoint unions of sets. A geometric instance dealing with disjoint unions of convex sets in Rd was proved in a special case by Larman in 1968 and in the general case by Amenta in 1996. Also covered by the conjecture is a topological extension of Amenta's theorem obtained by Kalai and Meshulam in 2008.

Morris's proof of the generalized pigeonhole principle is extremely involved, and the validity of some of his arguments is open to dispute. In the present paper, the principle is placed on a sound basis and established in a relatively short and transparent manner. This includes a particular case, left open by Morris, which is applied here to families of disjoint unions of boxes in Rd.

2000 Mathematics Subject Classification 52A35 (primary), 05A18, 52A01 (secondary).


Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?




Disclaimer: Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department.