Let z be a convex function defined in a convex domain D of a finite-dimensional Euclidean space. Denote by z (n) the convolutions of z with elements of a d-type sequence of test functions and let ? z and ? z (n) be the measures of normal images corresponding to z and z (n). One of the main results of this work is that ? z (n)?? z in variation on a compact K ? D if and only if ? z is absolutely continuous on K with respect to Lebesgue measure. Bibliography: 9 titles.