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Aarhus Universitets segl

Closed form of the rotational Crofton formula

by Jérémy Auneau, Jan Rataj and Eva B. Vedel Jensen
CSGB Research Reports Number 2 (June 2010)

The closed form of a rotational version of the famous Crofton formula is derived. In the case where the sectioned object is a compact d-dimensional C2 manifold with boundary, the rotational average of intrinsic volumes measured on sections passing through a fixed point can be expressed as an integral over the boundary involving hypergeometric functions. In the more general case of a compact subset of Rd with positive reach, the rotational average also involves hypergeometric functions. For convex bodies, we show that the rotational average can be expressed as an integral with respect to a natural measure on supporting flats. It is an open question whether the rotational average of intrinsic volumes studied in the present paper can be expressed as a limit of polynomial rotation invariant valuations.

Keywords: Geometric measure theory, hypergeometric functions, integral geometry, intrinsic volume, stereology

2000 Mathematics Subject Classi cation: 60D05, 53C65, 52A22

Format available: PDF (3 MB)