geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, Kalai's 3^''d'' conjecture is a conjecture on the polyhedral combinatorics of

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 ...

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

s, made by

Gil Kalai Gil Kalai (born 1955) is the Henry and Manya Noskwith Professor Emeritus of Mathematics at the Hebrew University of Jerusalem, Israel, Professor of Computer Science at the Interdisciplinary Center, Herzliya, and adjunct Professor of mathematics ...

in 1989.. It states that every ''d''-dimensional centrally symmetric polytope has at least 3^''d'' nonempty

faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...

(including the polytope itself as a face but not including the empty set).

Examples

In two dimensions, the simplest centrally symmetric

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...

s 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 A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

, the

regular octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

, is also centrally symmetric, and has 6 vertices, 12 edges, 8 triangular sides, and 1 solid: . In higher dimensions, the hypercube , 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 In geometry, a Hanner polytope is a convex polytope constructed recursively by Cartesian product and polar dual operations. Hanner polytopes are named after Olof Hanner, who introduced them in 1956.. Construction The Hanner polytopes are construct ...

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

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 polytope In geometry, a simplicial polytope is a polytope whose facets are all simplices. For example, a ''simplicial polyhedron'' in three dimensions contains only triangular facesPolyhedra, Peter R. Cromwell, 1997. (p.341) and corresponds via Steinitz's ...

s: 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 polytopes of

split graph In graph theory, a branch of mathematics, a split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Split graphs were first studied by , and independently introduced by . A split graph may have m ...

s, 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