A support line and separation problem

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.

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)

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. So how can
I do this with shapes?

Please help. Thanks

.

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