Multiple packings and coverings of a sphere

We reveal some relations between multiple packings and coverings of the $(n-1)$-dimensional unit sphere in $E^n$, $n\ge 4$, by a given number of spherical caps. We give estimates of the radii of those caps and consider several extremal cases of multiple packings and coverings of the sphere. Basing on minimaximin models, we suggest algorithms of numerical optimization of multiple packings and coverings of the sphere. A criterion whether or not a point belongs to a convex polygon in $E^n$, $n\ge 2$, is suggested.