Linear inequalities system’s solutions least distant from origin of coordinates

The problem of searching the least distant point of polyhedron from origin of coordinates in several statements is considered. A polyhedron is defined as a solution set of system of linear inequalities. Also the results of solving penalty functions minimizations problems including Holder norms with different power and weighting coefficients are considered. The multicriterion problem of searching vector of solutions of system of inequalitues with Pareto-minimal absolute values of all components is discussed. Theorems about relationship of sets of solutions of different statements of problem under consideration are formulated and proved.