Re: A support line and separation problem

On Fri, 24 Mar 2006 22:56:23 -0500, quasi <quasi@xxxxxxxx> wrote:

On 24 Mar 2006 18:17:55 -0800, "MK" <martagawel@xxxxxxxxxxx> wrote:

Hey everyone.

I have 2 questions:

1.) Find an example of a closed, non-convex set in R2 such that every
boundary point has a support line.

It's sort of a trick question

Hint #1: No one said the non-convex set had to be connected.

Hint #2: Start small.

You can even get a connected example.

Hint 3: Start with a closed convex polygonal region in the plane. Then
remove something small.

I've been playing with several shapes in 2-D that aren't convex
but just can't get it because the support lines cut through my shape in
the concave section of my shapes. Any ideas?

2.) Prove that the triangle inequality doesn't hold for sets. ie) find
sets A,B,C in R2 (2-D) so that
d(A,B) + d(B,C) < d(A,C)

Hint: Take 3 disks in R^2, and fool with the positions.

I think I'm having problems with this because I don't understand what
kind of sets I'm looking for. Plus, I have the actual triangle
inequality stuck in my head when I start drawing things out.

If you draw your sets as single points, of course the triangle
inequality will appear to hold.

So how can I do this with shapes?

Give your sets some territory.

And as suggested above, try simple shapes first.

As a general guideline, when you need to find counterexamples, start
with the simplest potential candidates, proceeding to more complicated
constructions only if forced.

quasi

quasi
.

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