Prove intersection of convex cones is convex.

[jvt05]

1. Let A and B be convex cones in a real vector space V. Show that A$\bigcap$B and A + B are also convex cones.

 

[Dick]

First state the definition of a convex cone, and then try to prove it. People will help if you show some effort first.

 

[jvt05]

well set C is a convex cone if for any x,y $\in$ C and any scalars a≥0, b≥0, ax + by $\in$ C

so let A and B be convex cones.
A$\bigcap$B would contain all elements x $\in$ both A and B.
This is where I am having trouble.

 

[jvt05]

other properties of convex cones:

1. for any positive scalar α and any x $\in$ C, the vector αx = (α/2)x + (α/2)x is in C.

2. set C is a convex cone if and only if αC = C and C + C = C.

perhaps my trouble is coming from the fact that I do not fully understand how these properties work.

 

[Dick]

well set C is a convex cone if for any x,y $\in$ C and any scalars a≥0, b≥0, ax + by $\in$ C

so let A and B be convex cones.
A$\bigcap$B would contain all elements x $\in$ both A and B.
This is where I am having trouble.

I'm having problems trying to follow your problem here. If x and y are in A then ax+by is in A, right? If x and y are in B then ax+by is in B, also right? If x and y are in AnB then they are in BOTH A and B. Doesn't that make ax+by in BOTH A and B? Hence an element of AnB?

 

[trenekas]

Hello. I don't want to create a new topic. I have very similar question about convex cone. I know what intersection and sum of two convex cones are also convex cone. But what's about union. The answer is that union of two convex cones may not be convex cone. But I can't understand why? Any thoughts? Thanks

 

[micromass]

Hello. I don't want to create a new topic. I have very similar question about convex cone. I know what intersection and sum of two convex cones are also convex cone. But what's about union. The answer is that union of two convex cones may not be convex cone. But I can't understand why? Any thoughts? Thanks

Can you find a counterexample??

Take two cones in real life, is their union a cone?

 

[trenekas]

OK. Thanks. But if one cone is subset of other then answer would be yes? I am right?

 

[micromass]

OK. Thanks. But if one cone is subset of other when answer would be yes? I am right?

Indeed.