MathDB

We call a bar of width

{w}

on the surface of the unit sphere

{\Bbb{S}^2}

, a spherical segment, centered at the origin, which has width

{w}

and is symmetric with respect to the origin. Prove that there exists a constant

{c>0}

, such that for any positive integer

{n}

the surface

{\Bbb{S}^2}

can be covered with

{n}

bars of the same width so that any point is contained in no more than

{c\sqrt{n}}

bars.

7