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# Simplex

Geometrically, simplices are:

• point,
• closed segment,
• closed triangle,
• closed pyramid, etc.

These are, respectively,

• $0$-dimensional simplex,
• $1$-dimensional simplex,
• $2$-dimensional simplex,
• $3$-dimensional simplex, etc.

The simplest example of $n$-simplex is the polygon in ${\bf R}^n$ with $n+1$ vertices at $(0,0,0,0,...,0), (1,0,0,0,...,0), (0,1,0,0,...,0), ..., (0,0,0,0,...0,1,0), (0,0,0,0,...,0,1)$.

A geometric $n$-simplex in ${\bf R}^{n+1}$ is defined as the convex hull (the set of all convex combinations) of $n+1$ points

${\rm conv}\{v_0, v_1, ..., v_n \}$
$v_1 - v_0, ..., v_n - v_0$ are linearly independent.

If we treat the simplex as a cell complex, its topology is very simple:

Theorem. The $n$-simplex is homeomorphic to the $n$-ball ${\bf B}^n$.

There is however an additional geometric structure; a simplex has faces.

Example. Suppose $a$ is a $1$-simplex

$a = {\rm \hspace{3pt} conv}\{v_0,v_1 \}$.

Then its faces are $v_0$ and $v_1$. They can be easily described algebraically. An arbitrary point in a is a convex combination of $v_0$ and $v_1$:

$a_0v_0 + a_1v_1$ with $a_0 + a_1 = 1.$

What about $v_0$ and $v_1$? They are convex combinations too but of a special kind:

$v_0 = a_0v_0 + a_1v_1$ with $a_0 = 1, a_1 = 0,$

$v_1 = a_0v_0 + a_1v_1$ with $a_0 = 0, a_1 = 1.$

Example. Suppose ${\tau}$ is a $2$-simplex

$$a = {\rm \hspace{3pt}conv}\{v_0,v_1,v_2 \}.$$

An arbitrary point in a is a convex combination of $v_0,v_1,v_2$:

$$a_0v_0 + a_1v_1 + a_2v_2 with a_0 + a_1 + a_2 = 1.$$

To find all $1$-faces set one of these coefficient equal to $0$:

$a = a_0v_0 + a_1v_1 + a_2v_2$ with $a_0 + a_1 + a_2 = 1$ and $a_2 = 0$,

$b = a_0v_0 + a_1v_1 + a_2v_2$ with $a_0 + a_1 + a_2 = 1$ and $a_1 = 0,$

$c = a_0v_0 + a_1v_1 + a_2v_2$ with $a_0 + a_1 + a_2 = 1$ and $a_0 = 0$.

So,

$$a,b,c < {\tau}.$$

To find all $0$-faces set two of these coefficient equal to $0$:

$$v_0 = 1 \cdot v_0 + 0 \cdot v_1 + 0 \cdot v_2, {\rm \hspace{3pt} etc}.$$

An abstract $n$-simplex is simply any finite set $A_0A_1...A_n$.

Exercise. Prove that the boundary of an $n$-simplex is homeomorphic to ${\bf S}^{n-1}$. Hint: put the sphere inside the simplex.

Instead of ${\bf R}^n$, one can carry out this construction in any vector space.