Inequality of different metrics for discrete Luxemburg norms in finite-dimensional spaces

An exact inequality of different metrics is obtained for discrete Luxemburg norms in a finite-dimensional space. As a consequence, using this inequality, an inequality of different metrics is proved for Luxemburg norms on functions for which there is an upper bound for the norm of a derivative in terms of the norm of the function, and an alternative proof is presented for S.M. Nikol'skii's inequality of different metrics for norms of a trigonometric polynomial in Orlicz spaces.