View source for Convex polyhedron

The [[Convex hull|convex hull]] of a finite number of points in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c0263101.png" />. Such a convex polyhedron is the bounded intersection of a finite number of closed half-spaces. An infinite convex polyhedron is the intersection of a finite number of closed half-spaces containing at least one ray; the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c0263102.png" /> is also conventionally considered to be a convex polyhedron. In this sense a convex polyhedron is the closed convex hull of a finite number of points and rays. The dimension of a convex polyhedron is the minimal dimension of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c0263103.png" /> that contains it.

A convex polyhedron is a special case of a [[Convex set|convex set]]. Being an intersection of half-spaces, a convex polyhedron is described by a system of linear inequalities and may be studied by algebraic tools. The methods of minimization of linear forms on a convex polyhedron form the subject of [[Linear programming|linear programming]].

A convex polyhedron has a finite number of faces (intersections of the convex polyhedron with the supporting hyperplanes). Each face of a convex polyhedron is a convex polyhedron of lower dimension. Faces of the faces are also faces of the original polyhedron. One-dimensional faces are known as edges; zero-dimensional faces are known as vertices. A bounded convex polyhedron is the convex hull of its vertices.

In the theory of convex surfaces (cf. [[Convex surface|Convex surface]]) the boundary of a convex polyhedron, and sometimes a part of such a boundary, is called a convex polyhedron [[#References|[1]]]. In the latter case one speaks of a convex polyhedron with boundary. In elementary geometry it is accepted to define a polyhedron as a figure composed of polygons in a special manner [[#References|[2]]], after which a convex polyhedron is defined as a polyhedron lying on one side of the plane passing through any one of its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c0263104.png" />-dimensional faces.

A bounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c0263105.png" />-dimensional convex polyhedron has not fewer than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c0263106.png" /> vertices. The simplest structure is that of a [[Simplex|simplex]], which has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c0263107.png" /> vertices. Any bounded convex polyhedron can be subdivided into simplices with adjacent common faces.

In the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c0263108.png" /> there are five regular convex polyhedra: the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. For their properties and their analogues see [[Regular polyhedra|Regular polyhedra]]; [[Semi-regular polyhedra|Semi-regular polyhedra]]. For convex polyhedra with special structural features see [[Isogons and isohedra|Isogons and isohedra]]; [[Zonohedron|Zonohedron]]. Special types of convex polyhedra — stereohedra, parallelohedra, planigons (cf. [[Stereohedron|Stereohedron]]; [[Parallelohedron|Parallelohedron]]; [[Planigon|Planigon]]) — are connected with regular subdivisions of space.

The possible types of structure of the network of faces of convex polyhedra have not been exhaustively studied. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c0263109.png" /> be the number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631010.png" />-dimensional faces of a bounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631011.png" />-dimensional convex polyhedron; then Euler's relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631012.png" /></td> </tr></table>

applies. It has a topological nature: It is valid for any subdivision of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631013.png" /> into simple cells. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631014.png" />, it is possible to find a convex polyhedron with such a network structure in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631015.png" /> for any connected network of faces on the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631016.png" /> which does not form dihedral and self-intersecting cells (Steinitz's theorem). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631017.png" />, the structure of the network of faces of a convex polyhedron is less arbitrary than the possible subdivisions of the sphere [[#References|[3]]]. Specific extremal problems, involving the structure of the network of faces, the number of edges or their overall length, etc., may be posed in the class of convex polyhedra [[#References|[4]]].

Approximation of convex bodies by convex polyhedra is a universal method of investigation. Many results in [[Mixed-volume theory|mixed-volume theory]], existence theorems, uniqueness theorems, stability theorems for convex surfaces with fixed parameters, and a geometric method of solving the [[Monge–Ampère equation|Monge–Ampère equation]] have been obtained in this way. The effectiveness of this method is due to the fact that a convex polyhedron is defined by a finite number of data; the general theorems can be simply formulated for convex polyhedra; synthetic methods of investigation are applicable to convex polyhedra.

A large part of the theory of convex polyhedra has been formulated in the context of the theory of surfaces [[#References|[1]]]. In the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631018.png" /> two bounded convex polyhedra with identical faces in the same order can be brought into congruence by a shift (Cauchy's theorem). In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631019.png" />, for certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631021.png" /> which satisfy the relationship

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631022.png" /></td> </tr></table>

there exists a unique, up to a shift, convex polyhedron with exterior unit normals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631023.png" /> to the faces, and with face areas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631024.png" /> (Minkowski's theorem). An evolvent of planar polygons, glued together so that the result is homeomorphic to the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631025.png" /> and such that the sum of the angles pieced together around each vertex is smaller than or equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631026.png" />, is isometric to a convex polyhedron in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631027.png" />, and this convex polyhedron is unique up to a shift (Aleksandrov's theorem). Two convex polyhedra in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631028.png" /> can be brought into congruence by a shift if for every normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631029.png" /> the faces (with exterior normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631030.png" />) cannot be brought by shifting into a position such that one of this faces is part of the other, but does not coincide with it. For rays <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631031.png" /> issuing from a point and for numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631032.png" /> there exists a unique, up to a homothety, convex polyhedron with vertices on the rays <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631033.png" /> and curvatures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631034.png" /> at these vertices.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.D. Aleksandrov,   "Konvexe Polyeder" , Akademie Verlag  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Fejes Toth,   "Lagerungen in der Ebene, auf der Kugel und im Raum" , Springer  (1972)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.S. Alexandroff [P.S. Aleksandrov] (ed.)  et al. (ed.) , ''Enzyklopaedie der Elementarmathematik'' , '''4. Geometrie''' , Deutsch. Verlag Wissenschaft.  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B. Grünbaum,   "Convex polytopes" , Interscience  (1967)</TD></TR></table>



====Comments====
Nowadays the phrase convex polytope is more often used to describe the convex hull of finitely many points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026310/c02631035.png" />. A convex polyhedron is then the boundary of a convex polytope (cf. the first line of the fourth allinea in the article above). The intersection of finitely many half-spaces is called a polyhedral set; it is not necessarily bounded.