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Kalai's 3^d conjecture
In geometry, Kalai's 3''d'' 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''d'' nonempty faces (including the polytope itself as a face but not including the empty set).
==Examples==
In two dimensions, the simplest centrally symmetric convex polygons are the parallelograms, which have four vertices, four edges, and one polygon; . A cube is centrally symmetric, and has 8 vertices, 12 edges, 6 square sides, and 1 solid; . Another three-dimensional convex polyhedron, the regular octahedron, is also centrally symmetric, and has 6 vertices, 12 edges, 8 triangular sides, and 1 solid; .
In higher dimensions, the hypercube ()''d'' has exactly 3''d'' 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''d'' faces. If Kalai's conjecture is true, these polytopes would be among the centrally symmetric polytopes with the fewest possible faces.〔
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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