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An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085320/s0853201.png" />-dimensional polytope (cf. [[Polyhedron|Polyhedron]]) that is the convex hull of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085320/s0853202.png" /> points (the vertices of the simplex) which do not lie in any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085320/s0853203.png" />-dimensional plane. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085320/s0853204.png" /> or 3, the simplex is a point, an interval, a triangle, or a tetrahedron, respectively. The faces of a simplex are simplices of lower dimension. Two simplices of the same dimension are affinely equivalent. Every point of a simplex corresponds to a unique way of distributing a unit mass among its vertices in such a way that the centre of gravity is at the given point. This is used to introduce [[Barycentric coordinates|barycentric coordinates]] in a simplex and also serves as a method of generalizing the notion of a simplex to the infinite-dimensional case (see [[Choquet simplex|Choquet simplex]]; [[Simplex (abstract)|Simplex (abstract)]]). A simplex can be ascribed one of two orientations, which then induces a specific orientation on each of its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085320/s0853205.png" />-dimensional faces.
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An  $  n $-
 +
dimensional polytope (cf. [[Polyhedron|Polyhedron]]) that is the convex hull of  $  n + 1 $
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points (the vertices of the simplex) which do not lie in any  $  ( n - 1) $-
 +
dimensional plane. When  $  n = 0, 1, 2, $
 +
or 3, the simplex is a point, an interval, a triangle, or a tetrahedron, respectively. The faces of a simplex are simplices of lower dimension. Two simplices of the same dimension are affinely equivalent. Every point of a simplex corresponds to a unique way of distributing a unit mass among its vertices in such a way that the centre of gravity is at the given point. This is used to introduce [[Barycentric coordinates|barycentric coordinates]] in a simplex and also serves as a method of generalizing the notion of a simplex to the infinite-dimensional case (see [[Choquet simplex|Choquet simplex]]; [[Simplex (abstract)|Simplex (abstract)]]). A simplex can be ascribed one of two orientations, which then induces a specific orientation on each of its  $  ( n - 1) $-
 +
dimensional faces.
  
 
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====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Grünbaum,  "Convex polytopes" , Wiley  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. McMullen,  G.C. Shephard,  "Convex polytopes and the upper bound conjecture" , Cambridge Univ. Press  (1971)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Grünbaum,  "Convex polytopes" , Wiley  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. McMullen,  G.C. Shephard,  "Convex polytopes and the upper bound conjecture" , Cambridge Univ. Press  (1971)</TD></TR></table>

Latest revision as of 08:14, 6 June 2020


An $ n $- dimensional polytope (cf. Polyhedron) that is the convex hull of $ n + 1 $ points (the vertices of the simplex) which do not lie in any $ ( n - 1) $- dimensional plane. When $ n = 0, 1, 2, $ or 3, the simplex is a point, an interval, a triangle, or a tetrahedron, respectively. The faces of a simplex are simplices of lower dimension. Two simplices of the same dimension are affinely equivalent. Every point of a simplex corresponds to a unique way of distributing a unit mass among its vertices in such a way that the centre of gravity is at the given point. This is used to introduce barycentric coordinates in a simplex and also serves as a method of generalizing the notion of a simplex to the infinite-dimensional case (see Choquet simplex; Simplex (abstract)). A simplex can be ascribed one of two orientations, which then induces a specific orientation on each of its $ ( n - 1) $- dimensional faces.

Comments

References

[a1] B. Grünbaum, "Convex polytopes" , Wiley (1967)
[a2] P. McMullen, G.C. Shephard, "Convex polytopes and the upper bound conjecture" , Cambridge Univ. Press (1971)
How to Cite This Entry:
Simplex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simplex&oldid=48707
This article was adapted from an original article by V.A. Zalgaller (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article