Distance between two sets A and B in Rn

[waternight]

Homework Statement

A is a sequentially compact set, and B is a point ⃗v in Rn−A.
define the distance between A and B as dist( ⃗u0 , ⃗v), where you showed
that ⃗u0∈ A exists such that dist( ⃗u0 , ⃗v)≤dist(⃗u, ⃗v) for all ⃗u in A.
a) Use this example to state a definition of the distance between two sets in general,
giving the largest class of sets for which your definition works.
b) Prove that your definition is well defined on the class of sets for which you say it
applies in a), and give counterexamples showing you can’t weaken the conditions in your
definition.
c) think of another definition of distance which applies to any two sets in Rn . Check that this second definition gives a reasonable answer for the distance between an open ball and its boundary. If you restrict this second definition to the class of sets considered in a) you should get the same answer as a).

Homework Equations

The Attempt at a Solution

I don't have any clue of writing the definition and the proof. What does the largest class of sets mean?

 

[mighty2000]

Can someone help me with this?

Thank you for your post. I am a scientist and I would be happy to help you with your question.

To start off, let's define what we mean by "distance between two sets". If we have two sets, A and B, in Rn, we can define the distance between them as the shortest distance between any point in A and any point in B. In other words, the distance between A and B is the minimum distance between any two points, one in A and one in B.

Now, to answer part a) of your question, the largest class of sets for which this definition works is the class of all non-empty sets in Rn. This is because any two non-empty sets in Rn will have at least one point in common, and therefore the distance between them will be well-defined.

For part b), we need to prove that this definition is well-defined on the class of all non-empty sets in Rn. To do this, we need to show that the distance between two sets, A and B, is the same regardless of which points we choose in A and B. To do this, we can assume that there are two points, ⃗u1 and ⃗u2, in A and two points, ⃗v1 and ⃗v2, in B such that dist(⃗u1, ⃗v1) < dist(⃗u2, ⃗v2). We can then show that this leads to a contradiction, which proves that our definition is well-defined.

For part c), another possible definition of distance between two sets is the Hausdorff distance. This is defined as the maximum distance between any two points, one in each set. This definition also works for any two sets in Rn, and when applied to the class of all non-empty sets, it gives the same answer as the definition in part a).

To check that this second definition gives a reasonable answer for the distance between an open ball and its boundary, we can consider the example of a 2-dimensional open ball with radius r and its boundary, which is a circle with radius r. Using the Hausdorff distance, we can see that the maximum distance between any two points, one in the ball and one on the boundary, is r. This makes sense, as the boundary is at a distance of r