# Definition: Simplex

Let $$\mathcal R=(\mathbb R,V_{\mathbb R},v)$$ be an $$n$$-dimensional affine space with $$V_{\mathbb R}$$ as the associated vector space over the field of real numbers $$\mathbb R$$. Let $$P_0,P_1,P_2\ldots,P_s$$ be $$s,\, (s\le n)$$ affinely independent points. An $$s$$-dimensional simplex is the affine space defined by the barycentric coordinates. $\sigma^s:=\{\lambda_0P_0+\cdots+\lambda_sP_s\,:\,\lambda_i\in \mathbb R, \, \lambda_i\ge 0,\,i=0,\ldots,s\,\wedge\, \lambda_0+\cdots+\lambda_s=1\}.$

Please note that the definition of the $$s$$-dimensional simplex is almost the same as the definition of the whole affine span of the $$s$$-dimensional hyperplane

$\{\lambda_0P_0+\cdots+\lambda_sP_s\,:\,\lambda_i\in \mathbb R, \,i=0,\ldots,s\,\wedge\, \lambda_0+\cdots+\lambda_s=1\},$

except the additional condition that all $$\lambda_i\in \mathbb R$$ must greater or equal $$0$$.

If we choose $$m+1,\,m\le s$$ points from the $$s+1$$ points $$P_0,P_1,P_2\ldots,P_s$$, then they also are affinely independent and form themselves a new simplex $$\sigma^m$$, which is an affine subspace of the simplex $$\sigma^s$$, called its $$m$$-face. The $$0$$-faces of $$\sigma^s$$ are also called its vertices, the $$1$$-faces are called edges and its $$2$$-faces are called facets.

The following figure demonstrates some simplices $$\sigma^0$$, $$\sigma^1$$, $$\sigma^2$$, and $$\sigma^3$$:

(c) bookofproofs own work

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### References

#### Bibliography

1. Reinhardt F., Soeder H.: "dtv-Atlas zur Mathematik", Deutsche Taschenbuch Verlag, 1994, 10th Edition