The paper deals with fundamental inequalities for preinvex functions. The result relating to preinvex functions on the invex set that satisfies condition C shows that such functions are convex on every generated line segment. As an effect of that convexity, the paper provides symmetric forms of the most important inequalities which can be applied to preinvex functions.

The paper studies the problem of cluster detection in noisy environment. The solution of this problem is based on the well known Expectation Maximization (EM) algorithm. By utilizing the Mahalanobis distance, and modifying the hidden variable, the rejection procedure is constructed so that it omits data from calculation of the current iteration step. Thus we construct the adaptive framework for solving the above problem. Several numerical examples are presented to illustrate the proposed algorithm.

The paper considers the solution properties of an overdetermined system of linear equations in a given norm. The problem is observed as a minimization of the corresponding functional of the errors. Presenting the main results of $p$ norm it is shown that the functional is convex. Following the convex properties we examine minimization properties showing that the problem possesses regression, scale, and affine equivariant properties. As an example we illustrated the problem of finding weighted mean and weighted median of the data.

The paper studies the application of convex functions in order to prove the existence of optimal solutions of an overdetermined system of linear equations. The study approaches the problem by using even convex functions instead of projections. The research also relies on some special properties of unbounded convex sets, and the lower level sets of continuous functions.