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A finite set of closed convex polytopes in a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073630/p0736301.png" /> which together with each polytope contains all its faces and is such that the intersection between the polytopes is either empty or is a face of each of them. An example of a polyhedral complex is the set of all vertices, edges and two-dimensional faces of the standard three-dimensional cube. One considers also complexes consisting of an infinite but locally finite family of polytopes. The concept of a polyhedral complex generalizes the concept of a geometric [[Simplicial complex|simplicial complex]]. The underlying space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073630/p0736302.png" /> of a polyhedral complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073630/p0736303.png" /> is the union of all polytopes entering into it and is itself an (abstract) polyhedron (cf. [[Polyhedron, abstract|Polyhedron, abstract]]). The number of polytopes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073630/p0736304.png" /> as a rule is less than the number of simplices in a triangulation. A polyhedral complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073630/p0736305.png" /> is called a [[Subdivision|subdivision]] of a complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073630/p0736306.png" /> if their underlying spaces coincide and if each polytope from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073630/p0736307.png" /> lies in a certain polytope from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073630/p0736308.png" />. A star-like subdivision of a complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073630/p0736309.png" /> with centre at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073630/p07363010.png" /> is obtained by means of a decomposition of the closed polytopes containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073630/p07363011.png" /> into cones with vertices at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073630/p07363012.png" /> over those faces that do not contain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073630/p07363013.png" />. Any polyhedral complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073630/p07363014.png" /> has a subdivision <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073630/p07363015.png" /> that is a geometric simplicial complex. Such a subdivision can be obtained without adding new vertices. It is sufficient, for example, to carry out in sequence the star-like subdivisions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073630/p07363016.png" /> with centres at all the vertices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073630/p07363017.png" />.
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A finite set of closed convex polytopes in a certain $\mathbf R^n$ which together with each polytope contains all its faces and is such that the intersection between the polytopes is either empty or is a face of each of them. An example of a polyhedral complex is the set of all vertices, edges and two-dimensional faces of the standard three-dimensional cube. One considers also complexes consisting of an infinite but locally finite family of polytopes. The concept of a polyhedral complex generalizes the concept of a geometric [[Simplicial complex|simplicial complex]]. The underlying space $|P|$ of a polyhedral complex $P$ is the union of all polytopes entering into it and is itself an (abstract) polyhedron (cf. [[Polyhedron, abstract|Polyhedron, abstract]]). The number of polytopes in $P$ as a rule is less than the number of simplices in a triangulation. A polyhedral complex $P_1$ is called a [[Subdivision|subdivision]] of a complex $P$ if their underlying spaces coincide and if each polytope from $P_1$ lies in a certain polytope from $P$. A star-like subdivision of a complex $P$ with centre at a point $a\in|P|$ is obtained by means of a decomposition of the closed polytopes containing $a$ into cones with vertices at $a$ over those faces that do not contain $a$. Any polyhedral complex $P$ has a subdivision $K$ that is a geometric simplicial complex. Such a subdivision can be obtained without adding new vertices. It is sufficient, for example, to carry out in sequence the star-like subdivisions of $P$ with centres at all the vertices of $P$.
  
 
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Latest revision as of 17:51, 11 April 2014

A finite set of closed convex polytopes in a certain $\mathbf R^n$ which together with each polytope contains all its faces and is such that the intersection between the polytopes is either empty or is a face of each of them. An example of a polyhedral complex is the set of all vertices, edges and two-dimensional faces of the standard three-dimensional cube. One considers also complexes consisting of an infinite but locally finite family of polytopes. The concept of a polyhedral complex generalizes the concept of a geometric simplicial complex. The underlying space $|P|$ of a polyhedral complex $P$ is the union of all polytopes entering into it and is itself an (abstract) polyhedron (cf. Polyhedron, abstract). The number of polytopes in $P$ as a rule is less than the number of simplices in a triangulation. A polyhedral complex $P_1$ is called a subdivision of a complex $P$ if their underlying spaces coincide and if each polytope from $P_1$ lies in a certain polytope from $P$. A star-like subdivision of a complex $P$ with centre at a point $a\in|P|$ is obtained by means of a decomposition of the closed polytopes containing $a$ into cones with vertices at $a$ over those faces that do not contain $a$. Any polyhedral complex $P$ has a subdivision $K$ that is a geometric simplicial complex. Such a subdivision can be obtained without adding new vertices. It is sufficient, for example, to carry out in sequence the star-like subdivisions of $P$ with centres at all the vertices of $P$.

References

[1] P.S. Aleksandrov, "Combinatorial topology" , Graylock , Rochester (1956) (Translated from Russian)


Comments

For extra references see also Polyhedral chain and Simplicial complex.

How to Cite This Entry:
Polyhedral complex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polyhedral_complex&oldid=31536
This article was adapted from an original article by S.V. Matveev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article