Ingeometry, aHanner polytope is aconvex polytope constructed recursively byCartesian product andpolar dual operations. Hanner polytopes are named after Swedish mathematicianOlof Hanner, who introduced them in 1956.[1]
The Hanner polytopes are constructed recursively by the following rules:[2]
They are exactly the polytopes that can be constructed using only these rules: that is, every Hanner polytope can be formed from line segments by a sequence of product and dual operations.[2]
Alternatively and equivalently to the polar dual operation, the Hanner polytopes may be constructed by Cartesian products anddirect sums, the dual of the Cartesian products. This direct sum operation combines two polytopes by placing them in two linearly independent subspaces of a larger space and then constructing theconvex hull of their union.[3][4]
Acube is a Hanner polytope, and can be constructed as a Cartesian product of three line segments. Its dual, theoctahedron, is also a Hanner polytope, the direct sum of three line segments. In three dimensions all Hanner polytopes are combinatorially equivalent to one of these two types of polytopes.[5] In higher dimensions thehypercubes andcross polytopes, analogues of the cube and octahedron, are again Hanner polytopes. However, more examples are possible. For instance, theoctahedral prism, a four-dimensionalprism with an octahedron as its base, is also a Hanner polytope, as is its dual, thecubical bipyramid.
Every Hanner polytope can be given vertex coordinates that are 0, 1, or −1.[6] More explicitly, ifP andQ are Hanner polytopes with coordinates in this form, then the coordinates of the vertices of the Cartesian product ofP andQ are formed by concatenating the coordinates of a vertex inP with the coordinates of a vertex inQ. The coordinates of the vertices of the direct sum ofP andQ are formed either by concatenating the coordinates of a vertex inP with a vector of zeros, or by concatenating a vector of zeros with the coordinates of a vertex inQ.
Because the polar dual of a Hanner polytope is another Hanner polytope, the Hanner polytopes have the property that both they and their duals have coordinates in{0,1,−1}.[6]
Every Hanner polytope iscentrally symmetric, and has exactly3d nonemptyfaces (including the polytope itself as a face but not including theempty set). For instance, the cube has 8 vertices, 12 edges, 6 squares, and 1 cube (itself) as faces;8 + 12 + 6 + 1 = 27 = 33. The Hanner polytopes form an important class of examples forKalai's 3d conjecture that all centrally symmetric polytopes have at least 3d nonempty faces.[3]
In a Hanner polytope, every two opposite facets are disjoint, and together include all of the vertices of the polytope, so that theconvex hull of the two facets is the whole polytope.[6][7] As a simple consequence of this fact, all facets of a Hanner polytope have the same number of vertices as each other (half the number of vertices of the whole polytope). However, the facets may not all be isomorphic to each other. For instance, in theoctahedral prism, two of the facets are octahedra, and the other eight facets aretriangular prisms. Dually, in every Hanner polytope, every two opposite vertices touchdisjoint sets of facets, and together touch all of the facets of the polytope.
TheMahler volume of a Hanner polytope (the product of its volume and the volume of its polar dual) is the same as for a cube or cross polytope. If theMahler conjecture is true, these polytopes are the minimizers of Mahler volume among all the centrally symmetricconvex bodies.[8]
The translates of ahypercube (or of anaffine transformation of it, aparallelotope) form aHelly family: every set of translates that have nonempty pairwise intersections has a nonempty intersection. Moreover, these are the onlyconvex bodies with this property.[9]For any other centrally symmetric convex polytopeK,Hanner (1956) definedI(K) to be the smallest number of translates ofK that do not form a Helly family (they intersect pairwise but have an empty intersection). He showed thatI(K) is either three or four, and gave the Hanner polytopes as examples of polytopes for which it is four.Hansen & Lima (1981) later showed that this property can be used to characterize the Hanner polytopes: they are (up to affine transformation) exactly the polytopes for whichI(K) > 3.[10]
The number of combinatorial types of Hanner polytopes of dimensiond is the same as the number ofsimpleseries–parallel graphs withd unlabeled edges.[4] Ford = 1, 2, 3, ... it is:
A more explicitbijection between the Hanner polytopes of dimensiond and thecographs withd vertices is given byReisner (1991). For this bijection, the Hanner polytopes are assumed to be represented geometrically using coordinates in{0,1,−1} rather than as combinatorial equivalence classes; in particular, there are two different geometric forms of a Hanner polytope even in two dimensions, the square with vertex coordinates(±1,±1) and the diamond with vertex coordinates(0,±1) and(±1,0). Given ad-dimensional polytope with vertex coordinates in{0,1,−1}, Reisner defines an associated graph whosed vertices correspond to the unit vectors of the space containing the polytope, and for which two vectors are connected by an edge if their sum lies outside the polytope. He observes that the graphs of Hanner polytopes are cographs, which he characterizes in two ways: the graphs with noinduced path of length three, and the graphs whose induced subgraphs are all either disconnected or the complements of disconnected graphs. Conversely, every cograph can be represented in this way by a Hanner polytope.[6]
The Hanner polytopes are theunit balls of a family of finite-dimensionalBanach spaces calledHanner spaces.[7] The Hanner spaces are the spaces that can be built up from one-dimensional spaces by and combinations.[1]