Octahedral and Euclidean projections of a point to a linear manifold

Many applied problems reduce to the general geometric problem of finding a point of a linear manifold in a finite-dimensional space that is closest to the origin. There are many specific formulations of this problem, including the search for octahedral and Euclidean projections, i.e., vectors of the linear manifold with smallest octahedral and Euclidean norms. We consider the properties of solutions to the problem of finding points of linear manifolds that are closest to the origin and relations between these solutions under various specifications of the problem. In particular, we study the properties of octahedral and Euclidean projections and analyze the influence on these projections of variation of weight coefficients in the norms.