Abstract
The purpose of this paper is to prove two integralgeometric formulae for convex bodies. Our results are expressed in terms of integrals with respect to the rigid-motion-invariant measure μ d , r on the space ℰ( d , r ) of all r -dimensional affine flats in d -dimensional Euclidean space E d . Rolf Schneider, in an unpublished note [6], has shown that for a convex polytope P in E d and 1 ≤ r ≤ d – 1 one has where η r ( P ) is the sum of the contents of the r -dimensional faces of P , η o ( E d–r ∩ P ) is the number of vertices of the ( d – r )-dimensional section E d – r ∩ P , and α( r ) is the content of the r -dimensional unit ball.
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