Distributing points on a sphere

The following question comes from a statement in Joshua Greene’s proof of the Kneser conjecture.

He states that, given nn and kk positive integers, we can find 2n+k2n+k points on Sk+1S^{k+1} such that no k+2k+2 points of these points lie on a great kk-sphere.

Why is this true?

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what is a great kk-sphere? the ones that have radius equal to the radius of the original sphere?
– Jorge Fernández Hidalgo
2 days ago

  

 

It is a subset of Sk+1⊂Rk+2S^{k+1} \subset \mathbb{R}^{k+2} obtained by taking one of the coordinates $x_1, \ldots, x_{k+2} to be zero.
– Mathmank
2 days ago

  

 

I don’t think that is correct, otherwise it would be trivial.
– Jorge Fernández Hidalgo
2 days ago

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Oh, right—the actual definition is the subset obtained by intersecting a hyperplane through the origin with our sphere.
– Mathmank
2 days ago

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1 Answer
1

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A great kk-sphere is the intersection of Sk+1S^{k+1} (as a subset of Rk+2\mathbb R^{k+2}) with a (k+1)(k+1)-dimensional hyperplane through the origin.
Such a hyperplane is determined by k+1k+1 linearly independent points.

Just start out with k+1k+1 linearly independent points, and repeatedly add
a point that is not on any of the hyperplanes determined by any k+1k+1 of the
points already present (always possible, since a finite union of hyperplanes will not cover the whole sphere).

  

 

Nice :){}{}{}{}
– Jorge Fernández Hidalgo
2 days ago