Abstract
In analogy to valuation characterizations and kinematic formulas of convex geometry, we develop a combinatorial theory of invariant valuations and kinematic formulas for finite lattices. Combinatorial kinematic formulas are shown to have application to some probabilistic questions, leading in turn to polynomial identities for Möbius functions and Whitney numbers.
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W. Blaschke, Vorlesungen über Integralgeometrie, 3rd Ed., VEB Deutsch. Verlag d. Wiss., Berlin, 1955. (Also: First Edition Part 1, 1935; Part II, 1937).
W.Y.C. Chen and G.-C. Rota,q-Analogs of the inclusion-exclusion principle, Discrete Math.104 (1992) 7–22.
S.S. Chern, On the kinematic formula in the euclidean space ofn dimensions, Amer. J. Math.74 (1952) 227–236.
J.H.G. Fu, Kinematic formulas in integral geometry, Indiana Univ. Math. J.39 (1990) 1115–1154.
S. Glasauer, Translative and kinematic integral formulae concerning the convex hull operation, Math. Z. (to appear).
J. Goldman and G.-C. Rota, On the fondations of combinatorial theory. IV. Finite vector spaces and Eulerian generating functions, Studies in Appl. Math.47 (1970) 239–258.
H. Hadwiger, Vorlesungen über Inhalt, Oberfläche, und Isoperimetrie, Springer Verlag, Berlin, 1957.
R. Howard, The kinematic formula in Riemannian homogeneous spaces, Memoirs of the A.M.S.509 (1993).
D. Klain, A short proof of Hadwiger's characterization theorem, Mathematika42 (1995) 329–339.
D. Klain, Kinematic formulas for finite vector spaces Discrete Math. (to appear).
D. Klain and G.-C. Rota, Introduction to Geometric Probability, Cambridge University Press, New York, 1997.
J. Kung, Ed., Gian-Carlo Rota on Combinatorics: Introductory Papers and Commentaries, Birkhäuser Verlag, Boston, 1995.
P. McMullen, Non-linear angle-sum relations for polyhedral cones and polytopes, Math. Proc. Camb. Phil. Soc.78 (1975) 247–261.
P. McMullen, Valuations and dissections, In: Handbook of Convex Geometry, Peter M. Gruber and Jörg M. Wills, Eds., North-Holland, Amsterdam, 1993, pp. 933–988.
P. McMullen and R. Schneider, Valuations on convex bodies, In: Convexity and its Applications, Peter M. Gruber and Jörg M. Wills, Eds., Birkhäuser Verlag, Boston, 1983, pp. 170–247.
N. Metropolis and G.-C. Rota, Combinatorial structure of the faces of then-cube, SIAM J. Appl. Math.35 (1978) 689–694.
N. Metropolis, G.-C. Rota, V. Strehl, and N. White, Partitions into chains of a class of partially ordered sets, Proc. Amer. Math. Soc.71 (1978) 193–196.
G.-C. Rota, On the foundations of combinatorial theory, I: Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie 2368 (1964) 340–368.
G.-C. Rota, On the combinatorics of the Euler characteristic, Studies in Pure Math., (Papers Presented to Richard Rado) Academic Press, London, 1971, pp. 221–233.
G.-C. Rota, Introduction to Geometric Probability, Lezioni Lincee held at the Scuola Normale Superiore Pisa, December 2–22, 1986.
L.A. Santaló, Integral Geometry and Geometric Probability, Reading, Addison-Wesley, MA, 1976.
R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge Unviesity Press, New York, 1993.
R. Schneider and J.A. Wieacker, Integral geometry, In: Handbook of Convex Geometry, Peter M. Gruber and Jörg M. Wills, Eds., North-Holland, Amsterdam, 1993, pp. 1349–1390.
R. Stanley, Enumerative Combinatorics, Wadsworth & Brooks/Cole Advanced Books and Software, Monterey, Calif., 1986.
J.H. van Lint and R.M. Wilson, A Course in Combinatorics, Cambridge University Press, New York, 1992.
W. Weil, Kinematic integral formulas for convex bodies, In: Contributions to Geometry, Proc. Geometry Symp., Siegen, 1978, J. Tölke and Jörg M. Wills, Eds., Birkhäuser Verlag, Boston, 1978, pp. 60–76.
G. Zhang, Dual kinematic formulas, Trans. Amer. Math. Soc. (to appear).
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Research supported in part by NSF grant #DMS 9626688.
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Klain, D.A. Kinematic formulas for finite lattices. Annals of Combinatorics 1, 353–366 (1997). https://doi.org/10.1007/BF02558486
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DOI: https://doi.org/10.1007/BF02558486