In geometry, Kalai's 3^''d'' conjecture is a

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...

on the

polyhedral combinatorics Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral comb ...

centrally symmetric In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...

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 In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...

Examples

In two dimensions, the simplest centrally symmetric convex polygons are the

parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...

s, which have four vertices, four edges, and one polygon: . A

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

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 In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ...

, 1sup>''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 , 1 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.

Generalizations

In the same work as the one in which the 3^''d'' conjecture appears, Kalai conjectured more strongly which the ''f''-vector of every convex centrally symmetric polytope ''P'' dominates the ''f''-vector of at least one Hanner polytope ''H'' of the same dimension. This means that, for every number ''i'' from 0 to the dimension of ''P'', the number of ''i''-dimensional faces of ''P'' is greater than or equal to the number of ''i''-dimensional faces of ''H''. If it were true, this would imply the truth of the 3^''d'' conjecture; however, the stronger conjecture was later disproven./

Status

The conjecture is known to be true for

d\le 4

. It is also known to be true for simplicial polytopes: it follows in this case from a conjecture of that every centrally symmetric simplicial polytope has at least as many faces of each dimension as the cross polytope, proven by . Indeed, these two previous papers were cited by Kalai as part of the basis for making his conjecture. Another special class of polytopes that the conjecture has been proven for are the

Hansen polytope Hansen may refer to: Places * Cape Hansen, Antarctica * Hansen, Idaho, town in the United States * Hansen, Nebraska, United States * Hansen, Wisconsin, town in the United States * Hansen Township, Ontario, Canada *Hansen, Germany, a small parish in ...

s of split graphs, which had been used by to disprove the stronger conjectures of Kalai.. The 3^''d'' conjecture remains open for arbitrary polytopes in higher dimensions.

References

{{DEFAULTSORT:Kalai's 3d conjecture Polyhedral combinatorics Conjectures Unsolved problems in geometry