Is the interior of an angle a convex set?

[LCharette]

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.

 

[Office_Shredder]

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

 

[LCharette]

Do you have any suggestions on how to prove the intersection of two half planes is convex?

 

[HallsofIvy]

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 ...

 

[blue_raver22]

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.