An update on the Hirsch conjecture

Details An update on the Hirsch conjecture An update on the Hirsch conjecture

2009-07-07

arxiv.org/abs/0907.1186v2

Jahresbericht der Deutschen Mathematiker-Vereinigung, Volume 112(2) (June 2010), 73-98

Mathematics

Combinatorics

28 pages, 6 figures. Many proofs have been taken out from version 2 and put into the appendix arXiv:0912.4235

Scientific paper

10.1365/s13291-010-0001-8.

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound $n-d$ is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.

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