📂Geometry

Definition of Simplex

Definition ¹

1. A $n$-simplex $\Delta^{n}$, whose convex hull consists of affinely independent $v_{0}, v_{1} , \cdots , v_{n} \in \mathbb{R}^{n+1}$, has vertices $v_{k}$. Formally, it is defined as follows: $$ \Delta^{n} := \left\{ \sum_{k} t_{k} v_{k} : v_{k} \in \mathbb{R}^{n+1} , t_{k} \ge 0 , \sum_{k} t_{k} = 1 \right\} $$
2. The faces of a $\Delta^{n}$ are the $n-1$-simplices $\Delta^{n-1}$ formed by removing a single vertex from $\Delta^{n}$. The boundary of $\Delta^{n}$ is the union of all its faces, denoted by $\partial \Delta^{n}$.
3. The interior of a simplex $\left( \Delta^{n} \right)^{\circ} := \Delta^{n} \setminus \partial \Delta^{n}$ is called an Open Simplex.

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• Affine independence means that $v_{1} - v_{0} , v_{2} - v_{0} , \cdots , v_{n} - v_{0}$ is linearly independent.

Description

A simplex is a concept encountered in areas such as linear programming and algebraic topology, characterized by its simplicity, as its name suggests. In Korean, it is colloquially termed 단체.

$n$-Simplex

The difference between a convex hull and a $n$-simplex, as per the definition, lies only in the affine independence of the given vectors. Unlike the convex hull of a set $X$, it is precisely represented by exactly $v_{0}, v_{1} , \cdots , v_{n}$, indicating it is a shape characterized solely by this representation.

$$ \left\{ \left( t_{0} , t_{1} , \cdots , t_{n} \right) \in \mathbb{R}^{n+1} : t_{k} \ge 0 , \sum_{k} t_{k} = 1 \right\} $$

This set is referred to as the Standard $n$-simplex. Considering only their combinations, regardless of the lengths of the vectors $v_{0}, v_{1} , \cdots , v_{n}$, it is aptly called standardization.

For example, consider $\Delta^{n} , n = 3,2,1,0$.

As seen, a $3$-simplex corresponds to a tetrahedron, a $2$-simplex to a triangle, a $1$-simplex to a line segment, and a $0$-simplex simply to a single point. The case where three points of a $2$-simplex lie on a single line is excluded by the assumption of affine independence. While a $n \ge 4$ cannot be geometrically represented, there is no issue in generalizing it.

Boundary and Open Simplex

Fundamentally, the concepts of boundary and open simplex are no different from those of boundary and interior discussed in metric spaces, even the notation is identical.

In the example, faces of the tetrahedron, a $3$-simplex, are shown as triangles, a $2$-simplex, verifying that the term “face” fits perfectly. Even the faces of a $1$-simplex, a line segment, are the endpoints, the $0$-simplices. It makes perfect sense to refer to this collection of faces as the boundary.

The boundary $\partial \Delta^{3}$ and open simplex $\left( \Delta^{3} \right)^{\circ}$ of $\Delta^{3}$ can be intuitively understood as shown in the picture.