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Closed form of the rotational Crofton formula
Author(s) -
Auneau Jérémy,
Rataj Jan,
Vedel Jensen Eva B.
Publication year - 2012
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201000028
Subject(s) - mathematics , hypergeometric function , mathematical analysis , boundary (topology) , regular polygon , invariant (physics) , rotation (mathematics) , measure (data warehouse) , pure mathematics , combinatorics , geometry , mathematical physics , database , computer science
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.