Convergence of Hölder projections to chebyshev projections

The problem of finding a point of a linear manifold with a minimal weighted Chebyshev norm is considered. In particular, to such a problem, the Chebyshev approximation is reduced. An algorithm that always produces a unique solution to this problem is presented. The algorithm consists in finding relatively internal points of optimal solutions of a finite sequence of linear programming problems. It is proved that the solution generated by this algorithm is the limit to which the Hölder projections of the origin of coordinates onto a linear manifold converge with infinitely increasing power index of the Hölder norms using the same weight coefficients as the Chebyshev norm.