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.