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In geometry, Kalai's 3 conjecture is a conjecture on the polyhedral combinatorics of centrally symmetric polytopes, made by Gil Kalai in 1989. It states that every d-dimensional centrally symmetric polytope has at least 3 nonempty faces (including the polytope itself as a face but not including the empty set).
Examples
For instance, the hypercube has exactly 3 faces, each of which can be determined by specifying, for each of the d coordinate axes, whether the face projects onto that axis onto the point 0, the point 1, or the interval . More generally, every Hanner polytope has exactly 3 faces. If Kalai's conjecture is true, these polytopes would be among the centrally symmetric polytopes with the fewest possible faces.
In the same work as the one in which the 3 conjecture appears, Kalai conjectured more strongly that the f-vector of every convex polytope dominates the f-vector of at least one Hanner polytope, but this was later disproven.
Status
The conjecture is known to be true for d ≤ 4. It remains open in higher dimensions.
References
- Kalai, Gil (1989), "The number of faces of centrally-symmetric polytopes", Graphs and Combinatorics, 5 (1): 389–391, doi:10.1007/BF01788696, MR 1554357.
- ^ Sanyal, Raman; Werner, Axel; Ziegler, Günter M. (2009), "On Kalai's conjectures concerning centrally symmetric polytopes", Discrete & Computational Geometry, 41 (2): 183–198, doi:10.1007/s00454-008-9104-8, MR 2471868/