Scientific journal
European Journal of Natural History
ISSN 2073-4972

THE INTERSECTION OF STRICTLY CONVEX SETS ON THE SPHERE OF SN

Gubajdullina N.A. 1 Khohlov A.G. 1
1 Tyumen State University
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We study convex sets M ≤ Sn, where Sn is an n-dimensional sphere.

The set M ≤ Sn is strictly convex [1] when it doesn’t contain diametrically opposite points of the sphere and with any pair of points it contains a small arc of a great or a certain (definable) circle.

We prove the following

Theorem. Let there exists the set of closed strictly convex sets gubaid01.wmf such that 1) gubaid02.wmf, 2) for all sets gubaid03.wmf s.t. gubaid04.wmf and gubaid05.wmf and for all natural numbers k satisfying conditions gubaid07.wmf minimal number of subsets gubaid08.wmf is equal to gubaid09.wmf, so maximal number of subsets A, containing k elements with the empty intersection is gubaid10.wmf.