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Face of face of a cone is a face. Proof?

  1. Dec 16, 2013 #1
    I could really do with some help. I'm trying to show that the face of a face of a convex polyhedral cone is again a face of that polyhedral cone. I have spent a couple hours thinking about this and CAN'T show it. The following apparently gives a proof of this, but it's surely invalid

    http://img30.imageshack.us/img30/4752/vsqc.png [Broken]

    The bit I have underlined. I can see literally no reason why [itex] \langle u, v \rangle \geq 0[/itex] would mean that [itex] \langle u, v \rangle = 0[/itex]. Can anyone help?
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Dec 17, 2013 #2

    Office_Shredder

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    This text is really hard to read because some stuff doesn't render or is mis type-setted, and they flip the meaning of v and w in the proof of 3. I believe the claim is using that v is contained in [itex] \check{\sigma}[/itex].

    We know that [itex] \left<v,w \right> [/itex] is non-negative because v is in [itex] \check{sigma}[/itex] and w is in [itex] \sigma[/itex]. Furthermore, p is non-negativve and [itex] \left<u,w \right> [/itex] is non-negative as well (for the same reason as [itex] \left<v,w\right> [/itex]. So we are adding two non-negative things together and getting zero. The only way this can occur is if both non-negative things were zero to begin with.
     
  4. Dec 17, 2013 #3

    ChrisVer

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    I would also try to answer that, but why is p non negative? except for if [itex]R_{+}[/itex] notation means positive reals... I interpreted it at first as the real numbers supplied by the action of summation.

    in the 2nd (3) and 2nd - it confused me more about it
     
  5. Dec 17, 2013 #4

    Office_Shredder

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    Chris, [itex] \mathbb{R}_+[/itex] does mean positive reals.
     
  6. Dec 17, 2013 #5
    Thanks for the replies, yeah I noticed the text was quite hard to read but it was the only proof I could find after a long search on google.

    I have actually figured it out (after harder searching) and did it before I saw this thread. I've attached in case anyone wants a look.

    http://img202.imageshack.us/img202/4387/2x7r.png [Broken]
     
    Last edited by a moderator: May 6, 2017
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