Let A be a set of points. Let B be the union of all segments of the form line segment PQ, where P and Q belong to A. Does it follow that B is convex? Why or why not?

Here's what I think:

Yes, it follows that B is convex because if B is contained in A and P, Q are members of A the line segment PQ is in B, then B has to belong to A and the line segment PQ lies in A making B convex.

Is this right? Can you help me if it isn't please?

My book defines that convex as for every two points P,Q of A the entire segment PQ lies in A.

Thanks for all of the help!

2. Originally Posted by rulergirl26
Let A be a set of points. Let B be the union of all segments of the form line segment PQ, where P and Q belong to A. Does it follow that B is convex? Why or why not?

Here's what I think:

Yes, it follows that B is convex because if B is contained in A and P, Q are members of A the line segment PQ is in B, then B has to belong to A and the line segment PQ lies in A making B convex.

Is this right? Can you help me if it isn't please?

My book defines that convex as for every two points P,Q of A the entire segment PQ lies in A.

Thanks for all of the help!
rulergirl26,

Hint:

Try points $\displaystyle A_1, A_2, A_3$ which are vertices of a triangle. Then B is the triangle, consisting of 3 line segments.

Is a triangle convex?

3. Originally Posted by awkward
rulergirl26,

Hint:

Try points $\displaystyle A_1, A_2, A_3$ which are vertices of a triangle. Then B is the triangle, consisting of 3 line segments.

Is a triangle convex?

Yes triangles are always convex because all line segments connecting two points on its surface lie entirely within the object.

So, the proof would start out as:

Given points $\displaystyle A_1, A_2, A_3$ which are vertices of a triangle. Then B is the triangle, consisting of 3 line segments...

But how or what do I say about the line segment PQ? Do I say that since th line segment PQ belongs to A then it has to lie in B making B convex? Something like that?

4. Originally Posted by rulergirl26
Yes triangles are always convex because all line segments connecting two points on its surface lie entirely within the object.

So, the proof would start out as:

Given points $\displaystyle A_1, A_2, A_3$ which are vertices of a triangle. Then B is the triangle, consisting of 3 line segments...

But how or what do I say about the line segment PQ? Do I say that since th line segment PQ belongs to A then it has to lie in B making B convex? Something like that?