Timeline for What is the largest subset of the sphere such that inner product of any two points in the set is nonnegative
Current License: CC BY-SA 4.0
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| Jun 7 at 22:00 | history | edited | Oscar Lanzi | CC BY-SA 4.0 |
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| Jan 19 at 13:03 | comment | added | RandomTensor | I guess there's not so much interest in this, but some of the randomly generated maximal sets in S^2 looked oblong, so they were missing a nice 3-symmetry of the positive orthant, or of the regular polygons described by @OscarLanzi . This probably isn't a worthwhile problem to work on, but I just thought it was interesting how it wasn't so nicely behaved. | |
| Jan 12 at 18:46 | history | edited | Oscar Lanzi | CC BY-SA 4.0 |
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| Jan 12 at 15:51 | history | edited | Oscar Lanzi | CC BY-SA 4.0 |
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| Jan 12 at 14:26 | history | edited | Daniel Weber | CC BY-SA 4.0 |
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| Jan 12 at 14:07 | comment | added | Oscar Lanzi | In three dimensions the cap beats all the polygons and looks like it's oprimal overall. Higher dimensions seem less sure, except in all cases the cap beats the orthant and it ultimatly isn't even close. | |
| Jan 12 at 14:04 | history | edited | Oscar Lanzi | CC BY-SA 4.0 |
Added more dimensions.
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| Jan 12 at 14:04 | comment | added | gerw | @OscarLanzi: Do you claim that the cap is the set with the largest area? I guess that it is. Further, it would be interesting to look at all sets which are maximal under inclusion and then find the one with the minimal area. I guess that it is the "triangle" (positive orthant). | |
| Jan 12 at 13:25 | comment | added | Oscar Lanzi | There are infitely many maximal sets. In particular, you can draw an equilateral triangle, regular pentagon, regular heptagon etc with longest chord = 90°; they are all maximal for their respective polygonal shapes and none fit inside each other. Nor do they fit inside the spherical cap described above, which beats all the polygons in area. | |
| Jan 12 at 11:47 | vote | accept | RandomTensor | ||
| Jan 12 at 11:46 | comment | added | RandomTensor | This is great, thanks! In case anyone is interested, I wrote a program that randomly sampled points from the from 2-sphere and added the points to a collection if its inner product with all points in the collection was positive. So it in some ways generated a random collection with this inner product condition. Its geomtery seemed to vary quite a bit. So I guess there are a lot of different maximal sets, as in you cannot add more volume without violating the inner product property, ignoring congruence via rotation. | |
| Jan 12 at 7:28 | comment | added | quarague | @SamHopkins One can still define a cap with angular radius 45° in a sphere of any dimension. The proof that <x,y> >=0 should be the same as in 3-d. Computing the volume of this cap is a little more complicated. Intuition on whether it remains bigger than the orthogonal cap is tricky. Proving optimality seems hard and I have no idea whether it is even true. | |
| Jan 12 at 1:33 | comment | added | Oscar Lanzi | Interesing question. Much harder to see even the fourth dimension! | |
| Jan 12 at 0:24 | comment | added | Sam Hopkins | This is interesting. Do you have any idea about what happens in general dimension? | |
| Jan 11 at 23:52 | history | edited | Oscar Lanzi | CC BY-SA 4.0 |
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| Jan 11 at 21:52 | history | edited | Oscar Lanzi | CC BY-SA 4.0 |
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| Jan 11 at 21:38 | history | edited | Oscar Lanzi | CC BY-SA 4.0 |
Added a generalization.
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| Jan 11 at 21:24 | history | answered | Oscar Lanzi | CC BY-SA 4.0 |