Some Properties of the Normal Image of Convex Functions

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 δ-type sequence of test functions and let νz and νz(n) be the measures of normal images corresponding to z and z(n). One of the main results of this work is that νz(n)→ νz in variation on a compact K ⊂ D if and only if νz is absolutely continuous on K with respect to Lebesgue measure. Bibliography: 9 titles.