Is the interior of an angle a convex set?

In summary, the speaker is trying to prove that the interior of <ABC is a convex set. They have defined the angle as the intersection of two half planes and have used the fact that each half plane is convex. However, they are stuck on proving that the intersection of two convex sets is also convex. They ask for suggestions and the conversation ends with a suggestion to use p and q in the intersection of two convex sets A and B.
  • #1
LCharette
9
0
I need to prove the interior of <ABC is a convex set. I know it is. I started by defining the angle as the intersection of two half planes and using the fact that each half plane is convex. I am stuck on where to go from here.
 
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  • #2
LCharette said:
I need to prove the interior of <ABC is a convex set. I know it is. I started by defining the angle as the intersection of two half planes and using the fact that each half plane is convex. I am stuck on where to go from here.

If you could prove the intersection of two convex sets is convex as well you'd be set
 
  • #3
Do you have any suggestions on how to prove the intersection of two half planes is convex?
 
  • #4
Let p and q be in the intersection of convex sets A and B.

p and q are both in A so ...

p and q are both in B so ...
 
  • #5


Yes, the interior of an angle is indeed a convex set. To prove this, we can start by defining an angle as the intersection of two half planes. Each half plane can be represented as a set of points that lie on one side of a line. Since a line is a straight path connecting two points, any two points within a half plane can be connected by a line segment that lies entirely within the half plane.

Now, let's consider any two points A and B within the interior of the angle <ABC. Since the interior of the angle is the intersection of two half planes, we know that both points A and B must lie within both half planes. This means that any line segment connecting A and B must also lie within both half planes.

Since each half plane is convex, we know that any line segment connecting two points within the half plane must also lie within the half plane. Therefore, the line segment connecting A and B must also lie entirely within the interior of <ABC.

This holds true for any two points A and B within the interior of <ABC, meaning that the interior of <ABC is a convex set. This is because any line segment connecting two points within the interior of <ABC must also lie entirely within the interior of <ABC.

In conclusion, the interior of an angle is a convex set because it can be represented as the intersection of two convex sets (the two half planes). This property holds true for any angle, making the interior of an angle a convex set.
 

Related to Is the interior of an angle a convex set?

1. What does it mean for a set to be convex?

For a set to be convex, it means that for any two points within the set, the line segment connecting them also lies within the set. In other words, the set is not "broken" and does not have any indentations or holes.

2. Why is it important for the interior of an angle to be a convex set?

The interior of an angle being a convex set is important because it ensures that the angle is well-defined and has a clear boundary. This is crucial in geometric and trigonometric calculations and proofs.

3. How can you prove that the interior of an angle is a convex set?

The interior of an angle can be proven to be convex by using the definition of convexity. We can show that for any two points within the interior of the angle, the line segment connecting them lies entirely within the interior of the angle.

4. Can the interior of an angle ever be a non-convex set?

No, the interior of an angle can never be a non-convex set. By definition, the interior of an angle is always a subset of the angle itself, which is a convex set. Therefore, the interior of an angle must also be a convex set.

5. How does the convexity of the interior of an angle relate to the convexity of the angle itself?

The convexity of the interior of an angle is directly related to the convexity of the angle itself. The interior of an angle being convex ensures that the angle itself is also convex, as the interior is a subset of the angle. Additionally, the convexity of the angle is dependent on the convexity of its interior, as any non-convexity in the interior would result in a non-convex angle.

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