Concentric circles in Q10
The inequality a^2 <= x1^2 + x2^2 <= b^2 in question 10 implies
1) that the points have to be outside of the smaller circle and inside the larger circle, and 2) that the circles have to be centered at the origin. Is that correct? 
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Re: Concentric circles in Q10
Thanks!

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Re: Concentric circles in Q10
In order to test the effective number of hypotheses in H, how can we fix the center of the circles? by fixing the center we restrict ourselves to one hypothesis. So, I am more confused by the last comment. My thought process is find a set of N points and look through all possible concentric circles, so all radii and centers, that will give me each dichotomy possible on N. And this is how H shatters N, not each single hypothesis. Is this correct?
What is a correct strategy to approach this problem? can we reduce it to 1D with an interior interval an exterior interval (to infinity) =1 and the 2 inbetween regions (between the 2 circles) =+1 then it becomes a more complex version of the 2.3 c, the positive, negative intervals? 
Re: Concentric circles in Q10
the last comment confused me a little bit.
For a given set of N points, we should change the center of the sphere to get as many dichotomies as we can, thus measuring the effective number of hypotheses (spheres) in this hypothesis set. Does it make sense to move project the spheres from 3D to 1D and look at the problem as intervals of +1 for a<=x<=b and a>=x>=b? 
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Thanks 
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Re: Concentric circles in Q10
When I skimmed this question first of all, I started thinking about annuli with arbitrary centres, which is an interesting more powerful hypothesis set.

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