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Sphere Intersections (Posted on 2015-11-11) Difficulty: 3 of 5


   K = H(P1,a1) ∩ H(P2,a2) ∩ ... ∩ H(Pn,an)


        ai ≥ ai+1 > 0        for i = 1, 2, ... , n-1.


        P1En  and

        PiH(P1,a1) ∩ H(P2,a2) ∩ ... ∩ H(Pi-1,ai-1)
                                   for i = 2, 3, ... , n.


     • En denotes Euclidean n-space. The set of n-tuples of
             real numbers.

     • If P ∈ En and r is a real number greater than zero, then
                      H(P,r) = { Q ∈ En | δ(P,Q) = r }.
       A "sphere" with center P and radius r.

     • δ(P,Q) denotes the distance between points P and Q in En.


     Is K empty ?

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Some Thoughts My thinking on it. | Comment 2 of 4 |
P(1) is all of Euclidean n-space.

Let's use terminology derived from n=3, to make things somewhat specific and in keeping with the title of the puzzle.

H(P(1),a(1)) is the surface of a sphere of radius a(1). Of course intersected with the entire Euclidean n-space, it's still the same spherical surface.

P(2) is the above-mentioned spherical surface.

P(3) is the intersection of that spherical surface with H(P(2),a(2)). But what is H(P(2),a(2))? The instructions define only an unsubscripted H.

As it stands right now, I don't see any constraint on the P that goes into the H formula.   But if we're talking about H(P(2),a(2)), I'd assume that P, for the delta function, is located on P(2), that is the spherical surface previously mentioned.

So the intersection of this new sphere with the previous sphere will be a circle (i.e., one dimension lower than the previous sphere).

Each successive subscript will reduce the dimensions by one. In our chosen n=3 language, after a circle comes two points, and then nothing.

So in n=3, P(1) is a sphere, P(2) is a circle and P(3) is two points. But the H function then reduces this to nothing, for the last intersection defining K, so K will in fact be empty.

There would be an alternative interpretation of the definition of H: that the P in the definition refers to the whole set of points in the preceding P, and therefore the H set includes all the points of distance r from any of the points in P. But this conflicts with the definition of delta, which I assume requires P to be a point rather than a set of points, and therefore a point chosen from among the points of the previous P.

  Posted by Charlie on 2015-11-11 15:28:50
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