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Convex set : characteristic cone 
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#1
Jun1511, 08:44 AM

P: 5

Hello :)
I have been giving a mathematical problem. But I find difficulties solving this. Therefore, I will be very grateful if anybody might wanted to help? The problem is "Let K be a compact convex set in R^n and C a closed convex cone in R^n. Show that ccone (K + C) = C."  Julie. 


#2
Jun1511, 09:08 AM

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Hi wjulie and welcome to PF!
I'm not familiar with the terminology "characteristic cone", is it perhaps the same thing as a recession cone? ( http://planetmath.org/encyclopedia/D...ConvexSet.html ) It is obvious that [itex]C\subseteq ccone(K+C)[/itex]. Assume that this inclusion was strict, then there would be a direction d which is not in C. This d has a >0 distance from C. Thus the multiples of d grow further away from C. That is, the distance from d to C becomes arbitrarily big. But we still have that d is in ccone(K+C). Can you find a contradiction with that? 


#3
Jun1511, 09:34 AM

#4
Jun1511, 09:40 AM

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P: 18,331

Convex set : characteristic cone
I think you made a mistake in your picture since T and S are exactly the same there.
But I see what you mean. Let's prove this in steps. Let's begin with this: let d be a direction not in C. Can you prove that the distance between x+rd and C becomes arbitrarily large as r becomes large? I.e. can you show that [itex]d(x+rd,C)\rightarrow +\infty[/itex] as [itex]r\rightarrow +\infty[/itex]? 


#5
Jun1511, 10:03 AM

P: 5

hmm i can't quite see the trick. But K has no direction because it is compact?



#6
Jun1511, 10:12 AM

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P: 18,331

Do you see intuitively why it must be true?
Consider for example the cone [itex]C=\{(x,0)~\vert~x\in \mathbb{R}\}[/itex] in [itex]\mathbb{R}^2[/itex]. Take something not in C, for example (1,1). Do you see that multiples of (1,1) are getting further away from C? That is, if [itex]r\rightarrow +\infty[/itex], then the distance between (r,r) becomes arbitrarily large. The general case is quite the same... 


#7
Jun1511, 10:17 AM

P: 5

i can see the intuitive behind it now. But when i have shown that this distance grow larger, what's next? Where are we heading?



#8
Jun1511, 10:20 AM

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P: 18,331

Well, x+rd is getting further away from C. But if d is in ccone(K+C), it must hold that x+rd is in K+C. And thus we must be able to express x+rd=k+c. But as the distance between x+rd and c becomes large, then k must become large. Thus K must be unbounded.



#9
Jun1511, 10:22 AM

P: 5

Aha! I see. I got it now. Thank you, you have saved my day :)



#10
Jun1611, 06:11 AM

P: 2

"It is obvious that C⊆ccone(K+C)"
why is this obvious, please explain ? /Olga 


#12
Jun1611, 08:11 AM

P: 2

Isn't that just at proof of y+d belonging to the set (K+C), and not the characteristic cone(K+C)? 


#13
Jun1611, 08:21 AM

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P: 18,331

Well, to see that
[tex]C\subseteq ccone(C+K)[/tex] Take d in C, then for all x in C, we have that x+rd is in C. In particular rd is in C. Now, take c+k in C+K, then c+k+rd=k+(c+rd) is in C+K (since C is convex). Thus for every x in C+K, we have that c+rd is in C+K 


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