Why is it impossible to cover a sphere that has radius $R$ with $3$ open semispheres of radius $R$? In my mind I have the pictorial image of the situation, but I can't find a formal proof.
cover a sphere with 3 open semispheres
1
$\begingroup$
linear-algebra
geometry
-
0yes I mean open set. – 2012-11-10
1 Answers
4
The union of two such semi-spheres always leaves an antipodal pair uncovered (where their boundaries intersect). This pair cannot be covered by a third semi-sphere since that does not contain any antipodal pair.