Closed form of the rotational Crofton formula

Abstract

AbstractThe 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 (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 \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb R}^d$\end{document} 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.