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R. Schneider (1993)
Convex Bodies: The Brunn-Minkowski Theory
S. Chiu, D. Stoyan, W. Kendall, J. Mecke (1989)
Stochastic Geometry and Its Applications
R. Schneider (1993)
Convex Bodies: The Brunn–Minkowski Theory: Minkowski addition
R. Schneider, W. Weil (2000)
Stochastische Geometrie
K. Schladitz, S. Peters, D. Reinel-Bitzer, A. Wiegmann, J. Ohser (2006)
Design of acoustic trim based on geometric modeling and flow simulation for non-wovenComput. Mat. Sci., 38
Oleg Iliev, R. Lazarov, J. Willems (2007)
Numerical Study of Two-grid Preconditioners for 1-d Elliptic Problems with Highly Oscillating Discontinuous Coefficients, 7
R. Schneider, W. Weil (1986)
Translative and Kinematic Integral Formulae for Curvature MeasuresMathematische Nachrichten, 129
W. Weil (2001)
Mixed Measures and Functionals of Translative Integral GeometryMathematische Nachrichten, 223
R. Schneider, W. Weil (1992)
Integralgeometrie
P. Davy (1978)
Stereology - a statistical viewpointBulletin of the Australian Mathematical Society, 19
W. Weil (1990)
Iterations of translative integral formulae and non-isotropic Poisson processes of particlesMathematische Zeitschrift, 205
Hermann Fallert (1996)
Quermadichten fr Punktprozesse konvexer Krper und Boolesche ModelleMathematische Nachrichten
P. Goodey, W. Weil (1987)
Translative integral formulae for convex bodiesaequationes mathematicae, 34
W. Weil (1987)
Point Processes of Cylinders, Particles and FlatsActa Applicandae Mathematicae, 9
Translative integral formulas for curvature measures of convex bodies were obtained by Schneider and Weil by introducing mixed measures of convex bodies. These results can be extended to arbitrary closed convex sets since mixed measures are locally defined. Furthermore, iterated versions of these formulas due to Weil were used by Fallert to introduce quermass densities for (non-stationary and non-isotropic) Poisson processes of convex bodies and respective Boolean models. In the present paper, we first compute the special form of mixed measures of convex cylinders and prove a translative integral formula for them. After adapting some results for mixed measures of convex bodies to this setting we then use this integral formula to obtain quermass densities for (non-stationary and non-isotropic) Poisson processes of convex cylinders. Furthermore, quermass densities of Boolean models of convex cylinders are expressed in terms of mixed densities of the underlying Poisson process generalizing classical formulas by Davy and recent results by Spiess and Spodarev.
Acta Applicandae Mathematicae – Springer Journals
Published: Jul 22, 2008
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