# Thread: easy (?) intersection question

1. ## easy (?) intersection question

any help will be appreciated for this problem. I have an idea what the solution should be, but I'm unsure if I have enough justification.
--

Let p = (0,1) and q = (0,-1). Let X be the subset of $\displaystyle R^2$ that is formed by intersecting all the closed Euclidean balls that contain p and q. What is X?

Solution (so far):

We can immediately exclude closed Euclidean balls that do not have p and q as a boundary point, since there exist Euclidean balls contained in them, and that if X is contained in Y, the intersection is X.

With some basic geometry, I've shown that there are three balls that have p and q as boundary points: the unit ball centered at the origin, and balls of radius $\displaystyle sqrt(2)$ centered at (-1,0) and (1,0).

Now, I want to say that the intersection between the last two balls mentioned is X, but I cannot convince myself that is the case. Am I doing something wrong?

Let p = (0,1) and q = (0,-1). Let X be the subset of $\displaystyle R^2$ that is formed by intersecting all the closed Euclidean balls that contain p and q. What is X?
Look $\displaystyle d(p,q)=\sqrt{2}>0$ so let $\displaystyle \delta=\frac{d(p,q)}{4}$.

Here are two closed balls: $\displaystyle \overline{\mathcal{B}\left(p;\delta\right)}~\&~\ov erline{\mathcal{B}\left(q;\delta\right)}$.

Can they intersect?

3. d(p,q) = 2, but a simple mistake that doesn't lead to any trouble.

I understand the direction that you are taking, using a sufficiently small delta so that the two closed balls don't intersect, but they don't contain p and q.

4. You are right. I misread it as $\displaystyle p1,0)~\&~q0,-1)$.
But none the less there are two disjoint closed balls centered at p&q. Their intersection is empty.

5. Thank you for your help, but I'm still confused. If both Euclidean closed balls contain p and q, then the intersection of those balls would at the very least contain the points p and q.

6. You mean that these balls each contain both p & q .
Does the problem really say just that.

7. That is what was said: " intersecting all the closed Euclidean balls that contain p and q."

If it had said " intersecting all the closed Euclidean balls that contain p or q", then I would have interpreted it the way you did. In this case, the intersection is the Euclidean line segment having p and q as endpoints.

8. Oh! *facepalm* I apologize for the confusion. I interpreted that they contain p or q. I will ask for clarification on the question today. Thank you both.