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Journal cites, added 5 DOIs, added 2 PMCs, added 1 issue number using AWB (9513)

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In geometry, a '''Hanner polytope''' is a [[convex polytope]] constructed recursively by [[Cartesian product]] and [[Dual polyhedron|polar dual]] operations. Hanner polytopes are named after [[Olof Hanner]], who introduced them in 1956.<ref name="h56">{{citation
| last = Hanner | first = Olof
| journal = Mathematica Scandinavica
| mr = 0082696
| pages = 65–87
| title = Intersections of translates of convex bodies
| volume = 4
| year = 1956}}.</ref>

==Construction==
The Hanner polytopes are constructed recursively by the following rules:<ref>{{citation
| last = Freij | first = Ragnar
| publisher = Department of Mathematical Sciences, Chalmers Institute of Technology
| series = Ph.D. thesis
| title = Topics in algorithmic, enumerative and geometric combinatorics
| url = http://publications.lib.chalmers.se/records/fulltext/156428.pdf
| year = 2012}}.</ref>
*A line segment is a one-dimensional Hanner polytope
*The Cartesian product of every two Hanner polytopes is another Hanner polytope, whose dimension is the sum of the dimensions of the two given polytopes
*The dual of a Hanner polytope is another Hanner polytope of the same dimension.
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.

Alternatively and equivalently to the polar dual operation, the Hanner polytopes may be constructed by Cartesian products and [[direct sum]]s, 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 the [[convex hull]] of their union.<ref name="k89"/><ref name="swz"/>

==Examples==
[[File:Dual Cube-Octahedron.svg|thumb|The three-dimensional [[cube]] and its dual, the [[octahedron]], the two three-dimensional Hanner polytopes]]
[[File:Octahedral prism.png|thumb|[[Schlegel diagram]] of the [[octahedral prism]]]]
A [[cube]] is a Hanner polytope, and can be constructed as a Cartesian product of three line segments. Its dual, the [[octahedron]], 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.<ref>{{citation
| last = Kozachok | first = Marina
| contribution = Perfect prismatoids and the conjecture concerning with face numbers of centrally symmetric polytopes
| pages = 46–49
| publisher = P.G. Demidov Yaroslavl State University, International B.N. Delaunay Laboratory
| title = Yaroslavl International Conference "Discrete Geometry" dedicated to the centenary of A.D.Alexandrov (Yaroslavl, August 13-18, 2012)
| url = http://bsg.uniyar.ac.ru/sites/default/files/papers/Alexandrov2012Thesis.pdf#page=46
| year = 2012}}.</ref> In higher dimensions the [[hypercube]]s and [[cross polytope]]s, analogues of the cube and octahedron, are again Hanner polytopes. However, more examples are possible. For instance, the [[octahedral prism]], a four-dimensional [[Prism (geometry)|prism]] with an octahedron as its base is also a Hanner polytope, as is its dual, the double pyramid over a cube.

==Properties==
===Coordinate representation===
Every Hanner polytope can be given vertex coordinates that are 0, 1, or −1.<ref name="reisner">{{citation
| last = Reisner | first = S.
| doi = 10.1112/jlms/s2-43.1.137
| issue = 1
| journal = Journal of the London Mathematical Society
| mr = 1099093
| pages = 137–148
| series = Second Series
| title = Certain Banach spaces associated with graphs and CL-spaces with 1-unconditional bases
| volume = 43
| year = 1991}}.</ref> More explicitly, if ''P'' and ''Q'' are Hanner polytopes with coordinates in this form, then the coordinates of the vertices of the Cartesian product of ''P'' and ''Q'' are formed by concatenating the coordinates of a vertex in ''P'' with the coordinates of a vertex in ''Q''. The coordinates of the vertices of the direct sum of ''P'' and ''Q'' are formed either by concatenating the coordinates of a vertex in ''P'' with a vector of zeros, or by concatenating a vector of zeros with the coordinates of a vertex in ''Q''.

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}.<ref name="reisner"/>

===Number of faces===
Every Hanner polytope is [[Point reflection|centrally symmetric]], and has exactly 3''d'' nonempty [[Face lattice|faces]] (including the polytope itself as a face but not including the empty 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 for [[Kalai's 3^d conjecture|Kalai's 3''d'' conjecture]] that all centrally symmetric polytopes have at least 3''d'' nonempty faces.<ref name="k89">{{citation
| last = Kalai | first = Gil | authorlink = Gil Kalai
| doi = 10.1007/BF01788696
| issue = 1
| journal = Graphs and Combinatorics
| mr = 1554357
| pages = 389–391
| title = The number of faces of centrally-symmetric polytopes
| volume = 5
| year = 1989}}.</ref>

===Pairs of opposite facets and vertices===
In a Hanner polytope, every two opposite facets are disjoint, and together include all of the vertices of the polytope, so that the [[convex hull]] of the two facets is the whole polytope.<ref name="reisner"/><ref name="msdw">{{citation
| last1 = Martini | first1 = H.
| last2 = Swanepoel | first2 = K. J.
| last3 = de Wet | first3 = P. Oloff
| arxiv = 1108.5046
| doi = 10.1007/s10957-009-9552-1
| issue = 1
| journal = Journal of Optimization Theory and Applications
| mr = 2545946
| pages = 149–157
| title = Absorbing angles, Steiner minimal trees, and antipodality
| volume = 143
| year = 2009}}.</ref> 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 the [[octahedral prism]], two of the facets are octahedra, and the other eight facets are [[triangular prism]]s. Dually, in every Hanner polytope, every two opposite vertices touch disjoint sets of facets, and together touch all of the facets of the polytope.

===Mahler volume===
The [[Mahler 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 the [[Mahler conjecture]] is true, these polytopes are the minimizers of Mahler volume among all the centrally symmetric [[convex body|convex bodies]].<ref>{{citation
| last = Kim | first = Jaegil
| arxiv = 1212.2544
| title = Minimal volume product near Hanner polytopes
| year = 2012}}.</ref>

===Helly property===
The translates of a [[hypercube]] (or of an affine transformation of it, a [[parallelotope]]) form a [[Helly family]]: every set of translates that have nonempty pairwise intersections has a nonempty intersection. Moreover, these are the only [[convex body|convex bodies]] with this property.<ref>{{citation
| last = Sz.-Nagy | first = Béla
| journal = Acta Universitatis Szegediensis
| mr = 0065942
| pages = 169–177
| title = Ein Satz über Parallelverschiebungen konvexer Körper
| url = http://acta.fyx.hu/acta/showCustomerArticle.action?id=6292&dataObjectType=article
| volume = 15
| year = 1954}}.</ref>
For any other centrally symmetric convex polytope ''K'', {{harvtxt|Hanner|1956}} defined ''I''(''K'') to be the smallest number of translates of ''K'' that do not form a Helly family (they intersect pairwise but have an empty intersection). He showed that ''I''(''K'') is either three or four, and gave the Hanner polytopes as examples of polytopes for which it is four. {{harvtxt|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 which ''I''(''K'') > 3.<ref>{{citation
| last1 = Hansen | first1 = Allan B.
| last2 = Lima | first2 = Ȧsvald
| doi = 10.1007/BF02392457
| issue = 1-2
| journal = Acta Mathematica
| mr = 594626
| pages = 1–23
| title = The structure of finite-dimensional Banach spaces with the 3.2. intersection property
| volume = 146
| year = 1981}}.</ref>

==Combinatorial enumeration==
The number of combinatorial types of Hanner polytopes of dimension ''d'' is the same as the number of [[simple graph|simple]] [[series-parallel graph]]s with ''d'' unlabeled edges.<ref name="swz">{{citation
| last1 = Sanyal | first1 = Raman
| last2 = Werner | first2 = Axel
| last3 = Ziegler | first3 = Günter M. | author3-link = Günter M. Ziegler
| doi = 10.1007/s00454-008-9104-8
| issue = 2
| journal = Discrete & Computational Geometry
| mr = 2471868
| pages = 183–198
| title = On Kalai's conjectures concerning centrally symmetric polytopes
| volume = 41
| year = 2009}}/</ref> For ''d'' = 1, 2, 3, ... it is:
:1, 1, 2, 4, 8, 18, 40, 94, 224, 548, ... {{OEIS|id=A058387}}.

A more explicit [[bijection]] between the Hanner polytopes of dimension ''d'' and the [[cograph]]s with ''d'' vertices is given by {{harvtxt|Reisner|1991}}.<ref name="reisner"/> 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 a ''d''-dimensional polytope with vertex coordinates in {0,1,−1}, Reisner defines an associated graph whose ''d'' 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 no [[induced 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.

==Hanner spaces==
The Hanner polytopes are the [[unit ball]]s of a family of finite-dimensional [[Banach space]]s called '''Hanner spaces'''.<ref name="msdw"/> The Hanner spaces are the spaces that can be built up from one-dimensional spaces by <math>\ell_1</math> and <math>\ell_\infty</math> combinations.<ref name="h56"/>

==References==
{{reflist}}

[[Category:Polytopes]]

M7GpppX diphosphatase - Revision history

en>Rjwilmsi: Journal cites, added 5 DOIs, added 2 PMCs, added 1 issue number using AWB (9513)