, Volume 9, Issue 3, pp 299-306

On surface area measures of convex bodies

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Abstract

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 jL j is dense (in the weak topology) in the set of signed Borel measures on the unit sphere which have barycentre 0.