Extreme Points of the Set of Bistochastic Operators

Abstract

In this article, we study the structure of the set of all quadratic bistochastic operators, as well as a complete description of all the extreme points of the set of quadratic operators. A necessary and sufficient condition for the bistochasticity of a quadratic stochastic operator is found. An analogue of Birkhoff's theorem on the extreme points of a set of bistochastic matrices is studied. A sufficient condition is found for the extremity of bistochastic quadratic operator and in a two-dimensional simplex.