Abstract
This paper attempts to look at the interconnections existing between metrics, convexity, and integral geometry from the point of view of combinatorial integral geometry. Along with general expository material, some new concepts and results are presented, in particular the sin2-representations of breadth functions, translative versions of mean curvature integral, and the notion of 2-zonoids. The main aim is to apply these new ideas for a better understanding of the nature of zonoids.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Buffon, G. L. L.: Essai d’Arithmetique morale, Supplement a l’Histoire Naturelle, v. 4, Paris, 1977.
Ambartzumian, R. V.: Combinatorial Integral Geometry, Wiley, Chichester, New York, 1982.
Kendall, M. G. and Moran, P. A. P.: Geometrical Probability, Griffin, London, 1963.
Deltheil, R.: Probabilities Geometriques. Gauthier-Villars, Paris, 1926.
Blaschke, W.: Kreis und Kugel, de Gruyter, Berlin, 1956.
Blaschke, W.: Vorlesungen über Integralgeometrie I, II, Hamburger mathematische Einzelschriften, Teubner, Leipzig, 1936, 1937.
Sylvester, J. J.: ‘On a Funicular Solution of Buffon’s “Problem of the Needle” in its Most General Form’, Acta Math. 14 (1890), 185–205.
Ambartzumian, R. V.: ‘The Solution to the Buffon—Sylvester Problem in R 3’, Z. Wahrscheinlichkeitstheorie verw. Geb. 27 (1974), 53–74.
Harding, E. F.: ‘The Number of Partitions of a Set of N Points in K Dimensions Induced by Hyperplanes’, Proc. Edinburgh Math. Soc. II, 15 (1967), 285–289.
Watson, D.: ‘On Partitions of N Points’, Proc. Edinburgh Math. Soc. II, 16 (1969), 263–264.
Schläfli, L.: Theorie der vielfachen Kontinuität, Bern, 1952.
Ambartzumian, R. V.: ‘A Synopsis of Combinatorial Integral Geometry’, Adv. Math. 37 (1980), 1–15.
Ambartzumian, R. V.: ‘On Combinatorial Foundations of Integral Geometry’, Izvestia Acad. Sci. Armenian SSR, Mathematics (English: Soviet J. Contemporary Analysis) 16, 4 (1981), 285–291.
Santalo, L. A.: Integral Geometry and Geometric Probability, Addison-Wesley, Reading, Mass., 1976.
Ambartzumian, R. V.: ‘Factorization in Integral and Stochastic Geometry’, Teubner-Texte zur Math. v. 65, 1984, pp. 14–33.
Minkowski, H.: ‘Volumen und Oberfläche’, Math. Ann. 57 (1903), 447–495.
Stoyan, D. and Mecke, J.: Stochastische Geometrie, Akademie-Verlag, Berlin, 1983.
Schneider, R. and Weil, W.: ‘Zonoids and Related Topics’, in Convexity and its Applications, Birkhäuser, Basle, 1983, pp. 296–317.
Funk, P.: ‘Über Flächen mit lauter geschlossenen geodätischen Linien’, Math. Ann. 74 (1913), 278–300.
Matheron, G.: Random Sets and Integral Geometry, Wiley, New York, 1975.
Panina, G. J.: ‘Translation—invariant Measures and Convex Bodies in R 3’, Zapiski nauchnih seminarov LOMI 157, 1987.
Aramian, R. G.: ‘On Stochastic Approximation of Convex Bodies’, in print.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 D. Reidel Publishing Company
About this chapter
Cite this chapter
Ambartzumian, R.V. (1987). Combinatorial Integral Geometry, Metrics, and Zonoids. In: Ambartzumian, R.V. (eds) Stochastic and Integral Geometry. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3921-9_2
Download citation
DOI: https://doi.org/10.1007/978-94-009-3921-9_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-8239-6
Online ISBN: 978-94-009-3921-9
eBook Packages: Springer Book Archive