Either p or -p, but not both? Because you can certainly cover the whole sphere with such disks. On Fri, May 4, 2018 at 5:17 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Define an open disk on the unit sphere S^2 centered at the point c in S^2 to be a set of the form
D = D(c,p,r) = {p in S^2 | dist(p, c) < r}
where 0 < r <= pi/2 and dist is distance measured along the sphere. Every such D is contained in some hemisphere.
Puzzle: ------- Suppose we have a collection of disjoint open disks on the unit sphere. Is is possible that for every point p of S^2, either p or its antipodal point -p lies inside some disk of the collection?
—Dan
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