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August Cleaner
August Cleaner
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Your question asAs stated does not seem to be interesting. It is clear thatyour question admits an immediate answer because the extreme point structure of finite dimensional convex sets in ainfinite-dimensional Banach spacespaces is not related to the structure of the Banach space: for any such set we can find an affine (and thus, as allpreserving extreme structure) map into any other infinite-dimensional Banach spaces have the same collectionspace.

Comment of finite-dimensional convex sets (the example mentioned in the comment illustratesYoav Kallus is an illustration of this). 

On the other hand, there is a very interesting theory of extreme points of unit balls of Banach spaces, in which $c_0$ plays an important role. See, for example, the paper of Fonf on Polyhedral Banach spaces.

Possibly it is worthwhile to redesign your question.

Your question as stated does not seem to be interesting. It is clear that extreme point structure of finite dimensional convex sets in a Banach space is not related to the structure of the Banach space, as all infinite-dimensional Banach spaces have the same collection of finite-dimensional convex sets (the example mentioned in the comment illustrates this). On the other hand, there is a very interesting theory of extreme points of unit balls of Banach spaces, in which $c_0$ plays an important role. See, for example, the paper of Fonf on Polyhedral Banach spaces.

Possibly it is worthwhile to redesign your question.

As stated your question admits an immediate answer because the extreme point structure of finite dimensional convex sets in infinite-dimensional Banach spaces is not related to the structure of the Banach space: for any such set we can find an affine (and thus, preserving extreme structure) map into any other infinite-dimensional Banach space.

Comment of Yoav Kallus is an illustration of this. 

On the other hand, there is a very interesting theory of extreme points of unit balls of Banach spaces, in which $c_0$ plays an important role. See, for example, the paper of Fonf on Polyhedral Banach spaces.

Possibly it is worthwhile to redesign your question.

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August Cleaner
August Cleaner
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Your question as stated does not seem to be interesting. It is clear that extreme point structure of finite dimensional convex sets in a Banach space is not related withto the structure of the Banach space, andas all infinite-dimensional Banach spaces have the same collection of finite-dimensional convex sets (the example mentioned in the comment illustrates this). On the other hand, there is a very interesting theory of extreme points of unit balls of Banach spaces, in which $c_0$ plays an important role. See, for example, the paper of Fonf on Polyhedral Banach spaces.

I would suggest that you either delete orPossibly it is worthwhile to redesign your question.

Your question as stated does not seem to be interesting. It is clear that extreme point structure of finite dimensional convex sets in a Banach space is not related with the structure of the Banach space, and the example mentioned in the comment illustrates this. On the other hand, there is a very interesting theory of extreme points of unit balls of Banach spaces, in which $c_0$ plays an important role. See the paper of Fonf on Polyhedral Banach spaces.

I would suggest that you either delete or redesign your question.

Your question as stated does not seem to be interesting. It is clear that extreme point structure of finite dimensional convex sets in a Banach space is not related to the structure of the Banach space, as all infinite-dimensional Banach spaces have the same collection of finite-dimensional convex sets (the example mentioned in the comment illustrates this). On the other hand, there is a very interesting theory of extreme points of unit balls of Banach spaces, in which $c_0$ plays an important role. See, for example, the paper of Fonf on Polyhedral Banach spaces.

Possibly it is worthwhile to redesign your question.

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August Cleaner
August Cleaner
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Your question as stated lacks contentsdoes not seem to be interesting. It is clear that extreme point structure of finite dimensional convex sets in a Banach space has nothing to dois not related with the structure of the Banach space, and the example mentioned in the comment illustrates this. On the other hand, there is a very interesting theory of extreme points of unit balls of Banach spaces, in which $c_0$ plays an important role. See the paper of Fonf on Polyhedral Banach spaces.

I would suggest that you either delete or redesign your question.

Your question as stated lacks contents. It is clear that extreme point structure of finite dimensional convex sets in a Banach space has nothing to do with the structure of the Banach space, and the example mentioned in the comment illustrates this. On the other hand, there is a very interesting theory of extreme points of unit balls of Banach spaces, in which $c_0$ plays an important role. See the paper of Fonf on Polyhedral Banach spaces.

I would suggest that you either delete or redesign your question.

Your question as stated does not seem to be interesting. It is clear that extreme point structure of finite dimensional convex sets in a Banach space is not related with the structure of the Banach space, and the example mentioned in the comment illustrates this. On the other hand, there is a very interesting theory of extreme points of unit balls of Banach spaces, in which $c_0$ plays an important role. See the paper of Fonf on Polyhedral Banach spaces.

I would suggest that you either delete or redesign your question.

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August Cleaner
August Cleaner
  • 902
  • 4
  • 8