geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, more specifically in polytope theory, 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 or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

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 co ...

of centrally symmetric

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 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 or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...

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 made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

s are the

parallelogram In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...

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

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

is centrally symmetric, and has 8 vertices, 12 edges, 6 square sides, and 1 solid: . Another three-dimensional

convex polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...

, the

regular octahedron In geometry, a regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Regular octahedra occur in nature as crystal structures. An octahedron, more generally, can be any eight-sided polyh ...

, 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 ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...

, 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 Swedish mathematician Olof Hanner, who introduced them in 1956.. Construction The Hann ...

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.

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.

Related conjectures

In the same work as the one in which the 3^''d'' conjecture appears, Kalai conjectured more strongly that 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./ 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 A flag is a piece of fabric (most often rectangular) with distinctive colours and design. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design employed, and flags have ...

, that is, ''d''!2^''d'', among all centrally symmetric polytopes. Finally, both the 3^''d'' conjecture and the full flag conjecture are sometimes said to be combinatorial analogues of the Mahler conjecture. All three conjectures claim the Hanner polytopes to minimize certain combinatorial or geometric quantities, have been resolved in similar special cases, but are widely open in general. In particular, the full flag conjecture has been resolved in some special cases using geometric techniques.Faifman, D., Vernicos, C., & Walsh, C. (2023). Volume growth of Funk geometry and the flags of polytopes. arXiv preprint arXiv:2306.09268.

References

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