Affine Integral Geometry from a Differentiable Viewpoint
نویسندگان
چکیده
The notion of a homogeneous contour integral is introduced and used to construct affine integral invariants of convex bodies. Earlier descriptions of this construction rely on a Euclidean structure on the ambient vector space, so the behavior under linear transformations is established after the fact. The description presented here relies only on the linear structure and a Lebesgue measure on the ambient vector space, making its behavior under linear transformations more transparent. Also, recent work by Ludwig classifying different types of affine or linearly invariant valuations on convex bodies is reviewed. The invariants obtained using the homogeneous contour integral are exactly the invariants that arise in Ludwig’s classification theorems.
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