On surface area measures of convex bodies

The set L_j of jth-order surface area measures of convex bodies in d-space is well known for j=d–1. A characterization of L_j was obtained by Firey and Berg. The determination of L_j, for j{2, ..., d–2}, is an open problem. Here we show some properties of L_j concerning convexity, closeness, and size. Especially we prove that the difference set L_j–L_j is dense (in the weak topology) in the set of signed Borel measures on the unit sphere which have barycentre 0.