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jvt05
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1. Let A and B be convex cones in a real vector space V. Show that A[itex]\bigcap[/itex]B and A + B are also convex cones.
jvt05 said:well set C is a convex cone if for any x,y [itex]\in[/itex] C and any scalars a≥0, b≥0, ax + by [itex]\in[/itex] C
so let A and B be convex cones.
A[itex]\bigcap[/itex]B would contain all elements x [itex]\in[/itex] both A and B.
This is where I am having trouble.
trenekas said: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
trenekas said:OK. Thanks. But if one cone is subset of other when answer would be yes? I am right?
A convex cone is a set that contains all the points on a line segment connecting any two points within the set. This means that if you pick any two points within the set, all the points on the line connecting them will also be included in the set.
Proving that the intersection of convex cones is convex means showing that if you have two or more convex cones, the set of points that are contained in all of these cones is also a convex cone. In other words, the intersection of convex cones is also a convex cone.
Proving that the intersection of convex cones is convex is important because it helps us understand the behavior of convex cones. It also allows us to use convex cones in various mathematical and scientific applications with the assurance that their intersection will also be a convex cone.
To prove that the intersection of convex cones is convex, you can use the definition of a convex cone and show that the set of points contained in all of the given convex cones also satisfies this definition. This can be done by using mathematical proofs and properties of convex cones.
Yes, there are many real-world applications of proving the intersection of convex cones is convex. For example, in optimization problems, convex cones are often used to model constraints. By proving the intersection of these cones is convex, we can ensure that the solution space of the problem is also a convex set, making it easier to find an optimal solution. Convex cones are also used in economics, engineering, and statistics, among other fields.