# Simplex

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