Separation of two bounded point sets on the plane by a second order curve (Q2703336)

The author studies the problem of separation by a second order curve of two bounded point sets \(M\) and \(N\) on the plane separated by a hyperplane in the corresponding five-dimensional Euclidean space. NEWLINENEWLINENEWLINEThe following results are proved. If \(\text{conv } M\cap \text{conv } N\neq\emptyset\), \(\text{conv } M\cap N=\emptyset\) or \(\text{conv } N\cap M= \emptyset\), then there exists a separating second order curve of any kind (parabola, ellipse, hyperbola). If \(\text{conv } M\cap \text{conv } N\neq\emptyset, \text{conv } M\cap N\neq\emptyset\), \(\text{conv } N\cap M\neq\emptyset\), then there is only a separating hyperbola. If two bounded closed sets are separated by parabola, then there exists a separating ellipse. Existence of a separating ellipse in the case of the existence of a separating hyperbola is studied.