Date: 01 Jun 2005

Oriented matroid polytopes and polyhedral fans are signable

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While convex polytopes are well known to be shellable, an outstanding open question, of which the answer is likely to be negative, is whether the strictly larger classes of oriented matroid polytopes and polyhedral cone fans are also shellable. In this article we show in a unified way that both classes posses the somewhat weaker property of signability. In particular, this allows us to conclude that simplicial oriented matroid polytopes and fans are partitionable, and to prove they satisfy McMullen's upper bound theorem on the number of faces. We also discuss computational complexity aspects of signability and shellability, and pose questions regarding the hierarchy of signable and shellable complexes.

Research was supported by the Alexander von Humboldt Stiftung and by the Fund for the Promotion of Research at the Technion.