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Extra resources for Integral Geometry And Convexity: Proceedings of the International Conference, Wuhan, China, 18 - 23 October 2004

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We say that $ is equi-affine invariant if it is SL(n) invariant and translation invariant. These notions are important for real valued valuations. We say that a functional $ is SL(n) covariant if $(aK) = a$(K) WK € /C",Va G SL(n). This notion is important for vector and tensor valued valuations as well as for convex body and star body valued valuations. In the following, we describe classification theorems for SL(n) invariant and SL(n) covariant valuations and make some remarks on related results for rotation invariant and rotation covariant valuations.

D 23 4. Alesker multiplication The other major tool that Alesker's approach offers is the definition of a commutative graded product on a dense subspace of Val. 1. (Alesker 6 ) Given A,B £ tC, the product of the valuations HAI^B is defined by (»A • HB){C) := vol 2 " ( A c + (A x {0}) + ({0} x B)) where A c := {(c,c) : c G C} C R 2 n is the diagonal embedding of C. If G C SO(n) is a compact subgroup then this product may be extended by continuity to (MG • /xg)(C) := / / J vol 2 " (Ac + (gA x {0}) + ({0} x hB)) dgdh.

In the 1930s, Blaschke obtained the first classification of real valued valuations that are SL(n) invariant. This was greatly extended by Hadwiger in his famous classification of continuous and rigid motion invariant valuations. Theorem 1 (Hadwiger 2 6 ) . A functional $ : /Cn —> M. , Cn € 1R such that * ( / 0 = co V0(K) + . . n. Here VQ{K),. ,Vn{K) are the intrinsic volumes of K; Vn = V is the ordinary volume, Vn-\ is proportional to the surface area and VQ is the Euler Characteristic. The classical theory of valuations and their applications in integral geometry and geometric probability are described in the books and surveys 26 > 35 .

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