Rublyov B. Quadratic recognition of sets and investigation of smooth metrics

Theoretical basis and framework for construction of the Smallest Enclosing Ellipse (SEE), Smallest Enclosing Ellipsoid (SEEd), Largest Inscribed Triangle (LIT), and Largest Inscribed Simplex (LIS) are created. Algorithms for constructing SEE and LIT on a plane, SEEd and LIS in a finite-dimensional space, and their approximations are developed. The algorithms construct exact SEE and SEEd in a finite number of steps with no error accumulation during transition between the steps. A method of constructing Elliptical Discriminant Functions (EFD) and Spherical Discriminant Functions (SDF) in a finite number of steps is developed. It reduces the problem either to the quadratic programming problem or to the problem of constructing Linear Discriminant Functions (LDF). Two methods of constructing LDF are considered - geometric and reducing to the linear programming problem. For the latter, an algorithm that solves the problem in linear time is presented. The tangent metric is defined, its properties are studied,and convergence of sequences of figures in this metric is considered. It is shown that the tangent metric is smooth and majorizes (in the corresponding class of figures on a plane) the metric of continuously differentiable functions. Software that implements the algorithms presented in the dissertation are developed. Also, applications of the algorithms to some practical problems of the national economy are described along with the perspective of their future applications.