ИСТИНА

The talk deals with combinatorial properties of metric projection P_E of a compact connected Riemann two-dimensional manifold M^2 onto its finitely connected subset E consisting of k closed connected sets E_j called continents. The point x \in M^2 is said to be exceptional if P_E(x) contains points from no less three different continents E_j. Our aim is to estimate the number of the exceptional points in terms of k and type of the manifold M^2. Theorem. Let compact connected Riemannian two-dimensional manifold M^2 contain k>2 continents. If M^2 is homeomorphic to a connected sum of m real projective planes, the maximum possible number of exceptional points is equal to 2k + 2m - 4. If M^2 is homeomorphic to a connected sum of a sphere with g tori, the maximum possible number of exceptional points is equal to 2k + 4g - 4. Similar estimate is proved for finitely connected subsets $E$ of a normed plane. The same problem was investigated for manifolds and normed spaces of higher dimension. It is easy to prove that for any integer k> 0 on any Riemann manifold or normed space of dimension n > 2 there exist three continents, for which there are at least k exceptional points. Thus, it is unclear how this problem can be generalized to the case of manifolds and normed spaces of higher dimension.