# Proving x* as an Extreme Point of a Convex Set | Homework Question

• ploppers
In summary, the statement is that x* is an extreme point of a convex set S if and only if the set S without x* is also a convex set. This is because if x* is an extreme point, then every point in S\{x*} can be connected to a point in S through a straight line, making it convex. Conversely, if S\{x*} is convex, then x* must be an extreme point because there are no other points in the set that can be connected to it through a straight line.
ploppers

## Homework Statement

Let x* be an element of a convex set S. Show that x* is an extreme point of S if and only if the set S\{x*} is a convex set.

## Homework Equations

(1-λ)x1 + λx2 exists in the convex set

## The Attempt at a Solution

I'm not too sure what S\{x*}, I asssumed it was the same as S/{x*} which is S over {x*}
I have is S?{x*} is a convex set then
λ(K/x*) + (1-λ)(P/x*) is a convex set were K and P are in the convex set S.
[λ(k) + (1-λ)(P)]/x* is in S/{x*}, but I can't see how it must me an extreme point

Usually \ is read as "minus":
$$S \setminus \{ x^* \} = \{ s \in S \mid s \neq x^* \}$$

I suppose the statement is intuitive: you can only keep drawing straight lines between points, if the point you take out is on an "edge".

Ahh thanks, I should have looked up the notation haha!

## 1. What is the definition of an extreme point in a convex set?

An extreme point in a convex set is a point that cannot be expressed as a convex combination of any two other points in the set. In other words, it is a point that lies on the boundary of the convex set and cannot be reached by taking a straight line between any two other points in the set.

## 2. How do you prove that a point is an extreme point of a convex set?

To prove that a point, x*, is an extreme point of a convex set, we must show that x* cannot be written as a convex combination of any two other points in the set. This can be done by assuming that x* can be expressed as a convex combination of two other points, and then using the definition of convexity to show that this is not possible.

## 3. What is the importance of extreme points in convex sets?

Extreme points are important because they help to define the structure and geometry of a convex set. They are also useful in optimization problems, as they can be used to determine the optimal solution.

## 4. Can a convex set have more than one extreme point?

Yes, a convex set can have multiple extreme points. In fact, for a convex set with n dimensions, there can be up to n extreme points.

## 5. Is every point on the boundary of a convex set an extreme point?

No, not every point on the boundary of a convex set is an extreme point. Some points on the boundary may be a convex combination of two other points in the set, making them non-extreme points. However, all extreme points of a convex set must lie on the boundary.

• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
8
Views
1K
• Calculus and Beyond Homework Help
Replies
6
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• General Math
Replies
9
Views
715
• Linear and Abstract Algebra
Replies
21
Views
1K
• Calculus and Beyond Homework Help
Replies
8
Views
2K
• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
5
Views
961
• Calculus and Beyond Homework Help
Replies
14
Views
612