Faculté des sciences

## Valuations with Crofton formula and Finsler geometry

### In: Advances in Mathematics, 2007, vol. 210, no. 2, p. 733-753

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... Plus

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Summary
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.