Abstract:
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.
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