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

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.
  • #1
ploppers
15
0

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
 
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  • #2
Usually \ is read as "minus":
[tex]S \setminus \{ x^* \} = \{ s \in S \mid s \neq x^* \}[/tex]

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".
 
  • #3
Ahh thanks, I should have looked up the notation haha!
 

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

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.

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