Closed form of the rotational Crofton formula

✍ Scribed by Jérémy Auneau; Jan Rataj; Eva B. Vedel Jensen

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 205 KB
• Volume: 285
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.201000028

✦ Synopsis

Abstract

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 \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.