# Convex set : characteristic cone

• wjulie
In summary: So by definition, d is in ccone(C+K).In summary, the conversation discusses the problem of showing that C is equal to the characteristic cone of K+C, and provides a proof by contradiction. The conversation also covers the intuition behind the proof and clarifies the definition of a characteristic cone.
wjulie
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.

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/DirectionOfAConvexSet.html )

It is obvious that $C\subseteq ccone(K+C)$. 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?

Last edited by a moderator:
Thanks for your reply :) and yes, a characteristic cone is the same as a recession cone.

Then I must show that T = S

I can show that if a belongs to S, then a must belong to T as well.
Let a=y+d belong to C.
if y belongs to C, and because C $\subseteq$ C+K, then y must belong to C+K
Therefore a=y+d must belong to C+K.

But how about the other way? I'm finding it quite difficult.

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 $d(x+rd,C)\rightarrow +\infty$ as $r\rightarrow +\infty$?

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

Do you see intuitively why it must be true?
Consider for example the cone $C=\{(x,0)~\vert~x\in \mathbb{R}\}$ in $\mathbb{R}^2$. 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 $r\rightarrow +\infty$, then the distance between (r,r) becomes arbitrarily large.

The general case is quite the same...

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?

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.

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

"It is obvious that C⊆ccone(K+C)"

why is this obvious, please explain ?

/Olga

Olga-Dahl said:
"It is obvious that C⊆ccone(K+C)"

why is this obvious, please explain ?

/Olga

See post #3.

micromass said:
See post #3.

Well, that isn't a useable argument, in my opinion though...

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

Well, to see that

$$C\subseteq ccone(C+K)$$

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

## 1. What is a convex set?

A convex set is a set of points where every line segment joining any two points in the set lies entirely within the set. In other words, if you take any two points in a convex set, all the points on the line segment connecting them are also in the set.

## 2. What is the characteristic cone of a convex set?

The characteristic cone of a convex set is the set of all nonnegative linear combinations of vectors in the set. In other words, it is the set of all possible directions that can be formed by taking a linear combination of vectors in the set.

## 3. How is the characteristic cone related to the convex set?

The characteristic cone is a mathematical representation of the convex set. It captures the directional properties of the convex set and allows for the characterization of the set in terms of its extreme points and directions.

## 4. Why is the characteristic cone important in convex set analysis?

The characteristic cone is important because it allows for the development of powerful mathematical tools and techniques for analyzing convex sets. It provides a way to describe the convex set in terms of its geometric properties, making it easier to study and understand.

## 5. Can the characteristic cone be used to determine if a set is convex?

Yes, the characteristic cone can be used to determine if a set is convex. If the characteristic cone of a set contains all possible directions between any two points in the set, then the set is convex. If not, then the set is not convex.

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