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Closed form of the rotational Crofton formula. (English) Zbl 1235.60010

Summary: 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 \(C^{2}\) manifold with boundary, the rotational average of intrinsic volumes (total mean curvatures) 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 \(\mathbb R^d\) 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.

MSC:

60D05 Geometric probability and stochastic geometry
53C65 Integral geometry
52A22 Random convex sets and integral geometry (aspects of convex geometry)
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