We say that n+1 points x1, x2, …, xn+1 in a linear space L are "in general position" if they do

We say that n+1 points x1, x2, …, xn+1 in a linear space L...

We say that $n+1$ points $x_{1}, x_{2}, \ldots, x_{n+1}$ in a linear space $\mathrm{L}$ are "in general position" if they do not belong to any $(\mathrm{n}-1)$-dimensional subspace of $\mathrm{L}$. The convex hull of a set of $n+1$ points $x_{1}, \mathrm{x}, \ldots, x_{n+1}$ in general position is called an $\mathrm{n}$-dimensional simplex, and the points $x_{1}, x_{2}, \ldots$, $x_{n+1}$ themselves are called the vertices of the simplex. Describe the zerodimensional, one-dimensional, two-dimensional and three-dimensional simplexes in Euclidean three-space $R^{3}$. Prove that the simplex with vertices $\mathrm{x}_{1}, x_{2}, \ldots, x_{n+1}$ is the set of all points in $\mathrm{L}$ which can be represented in the form where $$ x=\sum_{k=1}^{n+1} \alpha_{k} x_{k} $$ $$ \alpha_{k} \geqslant 0, \quad \sum^{n+1} \alpha_{k}=1 $$ Problem 4. Show that if the points $x_{1}, x, \ldots, x_{n+1}$ are in general position, then so are any $\mathrm{k}+1(\mathrm{k}<\mathrm{n})$ of them. Comment. Hence the $\mathrm{k}+1$ points generate a k-dimensional simplex, called a $\mathrm{k}$-dimensionalface of the $\mathrm{n}$-dimensional simplex with vertices $x_{1}$, $x_{2}, \ldots, x_{n+1}$

We say that $n+1$ points $x_{1}, x_{2}, \ldots, x_{n+1}$ in a linear space $\mathrm{L}$ are "in general position" if they do not belong to any $(\mathrm{n}-1)$-dimensional subspace of $\mathrm{L}$. The convex hull of a set of $n+1$ points $x_{1}, \mathrm{x}, \ldots, x_{n+1}$ in general position is called an $\mathrm{n}$-dimensional simplex, and the points $x_{1}, x_{2}, \ldots$, $x_{n+1}$ themselves are called the vertices of the simplex. Describe the zerodimensional, one-dimensional, two-dimensional and three-dimensional simplexes in Euclidean three-space $R^{3}$. Prove that the simplex with vertices $\mathrm{x}_{1}, x_{2}, \ldots, x_{n+1}$ is the set of all points in $\mathrm{L}$ which can be represented in the form
where
$$
x=\sum_{k=1}^{n+1} \alpha_{k} x_{k}
$$
$$
\alpha_{k} \geqslant 0, \quad \sum^{n+1} \alpha_{k}=1
$$
Problem 4. Show that if the points $x_{1}, x, \ldots, x_{n+1}$ are in general position, then so are any $\mathrm{k}+1(\mathrm{k}<\mathrm{n})$ of them.

Comment. Hence the $\mathrm{k}+1$ points generate a k-dimensional simplex, called a $\mathrm{k}$-dimensionalface of the $\mathrm{n}$-dimensional simplex with vertices $x_{1}$, $x_{2}, \ldots, x_{n+1}$

Key Concepts

Faces of a Simplex

A face of a simplex is any convex hull of a subset of its vertices. More precisely, given an n-dimensional simplex, any k+1 vertices (with k < n) form a k-dimensional simplex known as a k-dimensional face. This concept is critical for understanding the hierarchical structure of simplexes and plays a key role in combinatorial geometry, where the relationships between faces of different dimensions are studied.

Simplex

A simplex is a generalization of the concept of a triangle or tetrahedron to arbitrary dimensions. An n-dimensional simplex is defined as the convex hull of n+1 points in general position. For example, in three-dimensional space, a 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex is a triangle, and a 3-simplex is a tetrahedron. The geometric properties of simplexes are foundational in fields such as topology, computational geometry, and optimization.

Barycentric Coordinates

Barycentric coordinates provide a way to represent any point within a simplex as a weighted average of the simplex’s vertices. The weights (or coordinates) are nonnegative and sum to one. This coordinate system is particularly useful because it not only guarantees that the point lies within the simplex, but also uniquely identifies the point in terms of the relative contribution of each vertex, offering a natural framework for interpolation and other applications.

Convex Hull

The convex hull of a set of points is the smallest convex set that contains those points. It is formed by taking all convex combinations of the points, where a convex combination is any linear combination with nonnegative coefficients that sum up to one. In geometric terms, the convex hull ‘fills in’ the shape defined by the vertices, making it a fundamental concept in understanding polyhedral sets.

General Position

In the context of affine or linear spaces, a collection of points is said to be in general position if no subset of these points lies entirely within a subspace of lower dimension than expected. This means that the points are affinely independent; in an n-dimensional space, any n+1 points in general position will not all lie in any (n–1)-dimensional subspace, ensuring that they define the full n-dimensional geometry of interest.

Transcript

00:03 We are given statements and we are to determine if they are true or false.

00:07 And to explain, so in part a, we're given the statement that a polytopic is the convex whole of a finite set of points.

00:24 This is true.

00:26 A polytope is the convex whole of a finite set of points.

00:32 In fact, this is true by definition of a polytope.

00:42 Now in part b, the statement is, let p be an extreme point of a convex set s.

00:53 If u and v lie in this convex set s, and p lies on the line segment between u and v, and p is not equal to u, then p must be equal to v.

01:12 This statement is true.

01:17 So to see why, consider the definition.

01:21 Of an extreme point.

01:25 So first of all, let s be a convex set.

01:29 A point p in the convex set s is an extreme point of s if p is not in the interior of any line segment that lies in s.

01:39 So we're told that p lies on a line segment which is inside s, since u and v are both in s and s is a convex set, and we're told that p is not the endpoint u.

01:58 Well, if p was not the point u, the endpoint v, then p would lie in the interior of a line segment that goes through s.

02:06 And so p would not be an extreme point, which would be a contradiction.

02:11 So this is also true by definition.

02:20 Now, in part c, the statement is, if s is a non -empty convex subset of rn, then s is the convex hull of its profile...

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