A new rotational integral formula for intrinsic volumes in space forms

A new rotational version of Crofton's formula is derived for the intrinsic volumes of a domain Y in a space form. More precisely, a functional is defined on the intersection between Y and a totally geodesic submanifold (plane) through a fixed point, such that the rotational average of this functional is equal to the intrinsic volumes of Y. Particular cases of interest in stereology are considered for the Euclidean case.

[1]  N Roberts,et al.  Unbiased volume estimation with coaxial sections: an application to the human bladder , 1993, Journal of microscopy.

[2]  Adrian Baddeley,et al.  Stereology for Statisticians , 2004 .

[3]  L. Cruz-Orive,et al.  A new expression for the density of totally geodesic submanifolds in space forms, with stereological applications , 2009 .

[4]  L M Cruz-Orive,et al.  Particle number can be estimated using a disector of unknown thickness: the selector. , 1987, Journal of microscopy.

[5]  B. Dundas,et al.  DIFFERENTIAL TOPOLOGY , 2002 .

[6]  O. Varga.  Integralgeometrie 3. Croftons Formeln für den Raum , 1936 .

[7]  R. Gardner.  Geometric Tomography: Parallel X-rays of planar convex bodies , 2006 .

[8]  Brian D. Ripley,et al.  Integral Geometry and Geometrical Probability , 1978, The Mathematical Gazette.

[9]  Eva B. Vedel Jensen,et al.  A rotational integral formula for intrinsic volumes , 2008, Adv. Appl. Math..

[10]  H. Gundersen,et al.  The optical rotator , 1997, Journal of microscopy.

[11]  De-lin Ren,et al.  Topics in Integral Geometry , 1994 .

[12]  H J Gundersen,et al.  The nucleator , 1988, Journal of microscopy.

[13]  K. Ball.  CONVEX BODIES: THE BRUNN–MINKOWSKI THEORY , 1994 .

[14]  Cruz Orive,et al.  Stereology: meeting point of integral geometry, probability, and statistics. In memory of Professor Luis A. Santaló (1911-2001) , 2002 .

[15]  L. Cruz-Orive.  A new stereological principle for test lines in three‐dimensional space , 2005, Journal of microscopy.