How do I show that a subset is closed and convex?

In summary, the conversation centered around showing that the subset T, which is the convex hull of five given vectors, is a closed and convex subset. To demonstrate this, the speaker suggested finding a general formula for an element of T using four independent parameters and using all five vectors. They also mentioned the definition of a convex hull and how it can be used to show that a set is closed by finding a convex combination of two points in T. The other speaker was unsure how to use this information and asked for further clarification. The expert suggested trying to write out all the details and checking if the vector p lies in the convex hull of T to determine if it is closed.
  • #1
FightingWizard
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We have a vector p = (0, 0, 2) in R^3 and we have the subset S = {xp where x >= 0} + T, where T is the convex hull of 5 vectors: (2,2,2), (4,2,2), (2,4,2), (4,4,6) and (2,2,10).
How do I show that the subset T is a closed and convex subset?

I know that a subset is called convex if it contains the line segment between any two of its points: (1-t)u + tv for every u and v in the subset. I've tried to take two of those 5 vectors and see if there contains a line segment, but so far, it doesn't make any sense. I hope that you can help me with this problem.
 
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  • #2
Start by writing a completely general formula for an element of T.
Hint: it will have four independent parameters and it will use all five vectors, not just two of them.
 
  • #3
andrewkirk said:
Start by writing a completely general formula for an element of T.
Hint: it will have four independent parameters and it will use all five vectors, not just two of them.

I don't understand "writing a completely general formula for an element of T". Can you explain what you mean by that?
 
  • #4
A formula with four parameters that can represent any element of T by choosing the values of the parameters that make the formula give that element.
More here.
 
  • #5
FightingWizard said:
We have a vector p = (0, 0, 2) in R^3 and we have the subset S = {xp where x >= 0} + T, where T is the convex hull of 5 vectors: (2,2,2), (4,2,2), (2,4,2), (4,4,6) and (2,2,10).
How do I show that the subset T is a closed and convex subset?

I know that a subset is called convex if it contains the line segment between any two of its points: (1-t)u + tv for every u and v in the subset. I've tried to take two of those 5 vectors and see if there contains a line segment, but so far, it doesn't make any sense. I hope that you can help me with this problem.

Do you know what a "convex hull" is?
 
  • #6
Ray Vickson said:
Do you know what a "convex hull" is?

Yes, the convex hull of a subset is the set of all convex linear combinations of elements from T, such that the coefficients sum to 1.
But I don't understand how to use this to show that the subset T is closed and convex.
 
  • #7
FightingWizard said:
Yes, the convex hull of a subset is the set of all convex linear combinations of elements from T, such that the coefficients sum to 1.
But I don't understand how to use this to show that the subset T is closed and convex.

Take two points ##x## and ##y## in ##T##. Each of ##x## and ##y## can be expressed as convex combinations of the five given points. For ##0 \leq \lambda \leq 1##, can you write ##\lambda x + (1-\lambda) y## as a convex combination of the five given points? Try it and see, by writing down all the details.

Next: what does it mean for a set to be closed? Can you show why the convex hull satisfies the closure-conditions?
 
  • #8
Ray Vickson said:
Take two points ##x## and ##y## in ##T##. Each of ##x## and ##y## can be expressed as convex combinations of the five given points. For ##0 \leq \lambda \leq 1##, can you write ##\lambda x + (1-\lambda) y## as a convex combination of the five given points? Try it and see, by writing down all the details.

Next: what does it mean for a set to be closed? Can you show why the convex hull satisfies the closure-conditions?

So I need to find the convex combination of the five given points and then check if the vector p lies in the convex hull of T, and if it does then I can use the definition of closure to see if it is closed. Is that correct?
 
  • #9
FightingWizard said:
So I need to find the convex combination of the five given points and then check if the vector p lies in the convex hull of T, and if it does then I can use the definition of closure to see if it is closed. Is that correct?

I cannot figure out what you are trying to say, but if you think that is what you need to do then go ahead and actually try it.
 

1. How do I show that a subset is closed and convex?

To show that a subset is closed and convex, you need to prove that it satisfies two conditions: (1) for any two points in the subset, the line segment connecting them is also contained in the subset, and (2) the complement of the subset is open. This can be done using mathematical definitions and properties of closed and convex sets.

2. What is a closed set?

A set is closed if it contains all its limit points. In other words, every convergent sequence in the set has a limit point that is also in the set. In terms of convexity, a closed set also means that the line segment connecting any two points in the set will also be contained in the set.

3. What is a convex set?

A set is convex if for any two points in the set, the line segment connecting them is also contained in the set. This means that any point on the line segment between the two points is also in the set. Geometrically, a convex set is one that has no indentations or "dents" in its shape.

4. How do I prove that a set is convex?

To prove that a set is convex, you can use the definition of convexity and the properties of convex sets. You will need to show that for any two points in the set, the line segment connecting them is also contained in the set. This can be done by using algebraic or geometric arguments.

5. Can a set be both open and closed?

Yes, a set can be both open and closed. This type of set is called a clopen set. An example is the set of all real numbers between 1 and 2, which includes both 1 and 2 and all the numbers in between. It is closed because it contains all its limit points, and it is open because it does not contain its boundary points (1 and 2).

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