Valuations with Crofton formula and Finsler geometry

Valuations admitting a smooth Crofton formula are studied using Geometric Measure Theory and Rumin's cohomology of contact manifolds. The main technical result is a current representation of a valuation with a smooth Crofton formula. A geometric interpretation of Alesker's product is given for such valuations. As a first application in Finsler geometry, a short proof of the theorem of Gelfand-Smirnov that Crofton densities are projective is derived. The Holmes-Thompson volumes in a projective Finsler space are studied. It is shown that they induce in a natural way valuations and that the Alesker product of the k-dimensional and the l-dimensional Holmes-Thompson valuation is the (k + l)-dimensional Holmes-Thompson valuation. (c) 2006 Elsevier Inc. All rights reserved.