By John Edward Campbell

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H. : Convolution of convex valuations. Geom. Dedicata 123 (2006), 153–169. [25] Bernig, A. and Fu, J. H. : Hermitian integral geometry. Ann. of Math. 173 (2011), 907–945. : On the curvatura integra in a Riemannian manifold. Ann. of Math. 46 (1945), 674–684. : Curvature measures. Trans. Amer. Math. Soc. 93 (1959), 418– 491. [28] Fu, J. H. : Curvature measures and generalized Morse theory, J. Differential Geom. 30 (1989), 619–642. [29] Fu, J. H. : Monge–Amp`ere functions, I. Indiana Univ. Math. J.

4. (1) Any smooth measure on X is a smooth valuation. 2. (2) The Euler characteristic χ is also a smooth valuation. This fact is less obvious. In the current approach, it is a reformulation of a version of the Gauss–Bonnet formula due to Chern [26], who has constructed μ and ω to represent the Euler characteristic; his construction depends on the choice of a Riemannian metric on X. (3) The next example is very typical for integral geometry. Let X = CPn be the complex projective space. Let C Gr denote the Grassmannian of all complex projective subspaces of CPn of a fixed complex dimension k.

The first general result in this direction is as follows. 31 ([1]). Let G be a compact subgroup of the orthogonal group. The space ValG is finite-dimensional if and only if G acts transitively on the unit sphere. 18, if G acts transitively on the sphere, then ValG ⊂ Valsm . This equips ValG with the product. Evidently, we have also McMullen’s decomposition n ValG = ValG i . i=0 Thus ValG becomes a finite-dimensional commutative associative graded algebra with unit. 10. Moreover, it was shown by Bernig [20] that for such G G all G-invariant valuations are even.

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