On uniform continuity of convex bodies with respect to measures in Banach spaces

Let μ be a probability measure on a separable Banach space X. A subset U⊂X is μ-continuous if μ(∂U)=0. In the paper the μ-continuity and uniform μ-continuity of convex bodies in X, especially of balls and half-spaces, is considered. The μ-continuity is interesting for study of the Glivenko-Cantelli theorem in Banach spaces. Answer to a question of F. Topsøe is given.