Kalai's 3^d conjecture, in geometry and polytope theory, concerns the polyhedral combinatorics of centrally symmetric polytopes. Formulated in 1989 by the mathematician Gil Kalai, it states that every d-dimensional centrally symmetric polytope has at least 3^d nonempty faces.
Facts
StatementEvery d dimensional centrally symmetric polytope has at least 3 to the power d nonempty faces, including the polytope itself as a face but not including the empty set. 1 Proposed Year Progress Toward ResolutionKalai's 3^d conjecture is known to be true for d less than or equal to 4. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Kalai's 3^d Conjecture, Wikipedia
Sources
1. Kalai's 3^d Conjecture, Wikipedia
Lead section
It states that every d-dimensional centrally symmetric polytope has at least 3d nonempty faces (including the polytope itself as a face but not including the empty set).
Introduction section, second sentence
made by Gil Kalai in 1989.
Related conjectures
If it were true, this would imply the truth of the 3d conjecture; however, the stronger conjecture was later disproven. A related conjecture also attributed to Kalai is known as the full flag conjecture and asserts that the cube (as well as each of the Hanner polytopes) has the maximal number of (full) flags, that is, d!2d, among all centrally symmetric polytopes.
- Lead section
In Branch: Geometry, Lead paragraph
In geometry, more specifically in polytope theory, Kalai's 3d conjecture is a conjecture on the polyhedral combinatorics of centrally symmetric polytopes, made by Gil Kalai in 1989.
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