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Kinematische Schnittmaße bei gegebener Schnittsituation in der Integralgeometrie
[article]

Robert Sowada, Universität Stuttgart, Universität Stuttgart

2004

We will now consider a well known integral geometrical formula to give an example for the problem that is underlying this work. This problem shall now be exemplified and motivated: Let C be a fixed compact d-dimensional manifold (with twice differentiable boundary) embedded into the d-dimensional euclidean space. Then the integral of the Euler characteristic of the intersection of C with a hyperplane---in respect of the invariant density of the manifold of all hyperplanes---is given by the
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... s given by the integral of the (d-2)-th mean curvature over the boundary of C. So this formula---normalised in a appropiate way---shows that the expectation of the Euler characteristic of the intersection of C with a random hyperplane is given---up to a constant factor which is independent of C---by this integral over the boundary of C. In this integral only (differential) geometrical properties of the boundary of C appear. This leads to the question how the corresponding distribution looks like, i.e. to ask for the kinematic measure of those hyperplanes whose intersection with C has a Euler characteristic of a fixed predetermined value. There are only very few results to the last problem. Furthermore these just focus on the euclidean plane. In 1890 J. J. Sylvester examined the two situations that lines intersect all sets or alternatively at least one set of a given finite union of disjoint convex sets. For up to three sets explicit formulas depending on the mutual position of the sets are given. For more sets a constructive method to gain such a formula is specified. In 1966 R. Sulanke studied nets of bounded convex curves with predetermined support of the probability distribution of its number of intersections with lines. For regular closed curves R. V. Ambarcumjan finally gave a formula for the probability that a line has exactly k points of intersections---but there is no complete proof given in his paper. Starting from this background in our diploma theses Annette Gauger and I calculated the kinematic measures of lines and circles respec [...]

doi:10.18419/opus-4730
fatcat:4dijagylcjcmbfj7xyfkfi3wra