Points on a sphere
Posted: Tue Sep 29, 2009 12:14 pmI'm not sure where else this might go, so if this is in the wrong category someone can move it.
Show that the sum of the squares of the distances between n points on a unit sphere is at most n^2, and determine when that value equals n^2.
I've been working on this using vectors, and the best I've been able to do is show that the sum is less than 2n(n-1) and that it is n^2 when the points form the vertices of a regular polygon inscribed in a great circle, but it appears that anything beyond that involves adding a ton of cosines/dot products for which I can discern no useful pattern. Is there another way to look at this problem?
Show that the sum of the squares of the distances between n points on a unit sphere is at most n^2, and determine when that value equals n^2.
I've been working on this using vectors, and the best I've been able to do is show that the sum is less than 2n(n-1) and that it is n^2 when the points form the vertices of a regular polygon inscribed in a great circle, but it appears that anything beyond that involves adding a ton of cosines/dot products for which I can discern no useful pattern. Is there another way to look at this problem?