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

  • #1
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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
andrewkirk
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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
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
andrewkirk
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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
Ray Vickson
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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.
Do you know what a "convex hull" is?
 
  • #6
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
Ray Vickson
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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
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
Ray Vickson
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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.
 

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