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Dual kinematic formulas

Author(s): Gaoyong Zhang
Journal: Trans. Amer. Math. Soc. 351 (1999), 985-995.
MSC (1991): Primary 52A22; Secondary 53C65, 60D05
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Abstract: We establish kinematic formulas for dual quermassintegrals of star bodies and for chord power integrals of convex bodies by using dual mixed volumes. These formulas are extensions of the fundamental kinematic formula involving quermassintegrals to the cases of dual quermassintegrals and chord power integrals. Applications to geometric probability are considered.

References:

[G1]
R.J. Gardner, Intersection bodies and the Busemann-Petty problem, Trans. Amer. Math. Soc. 342 (1994), 435-445. MR 94e:52008

[G2]
-, A positive answer to the Busemann-Petty problem in three dimensions, Annals of Math. 140 (1994), 435-447. MR 95i:52005

[G3]
-, On the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies, Bull. Amer. Math. Soc. 30 (1994), 222-226. MR 94m:52007

[G4]
-, Geometric tomography, Cambridge University Press, Cambridge, 1995. MR 96j:52006

[H]
R. Howard, The kinematic formula in Riemannian homogeneous spaces, Mem. Amer. Math. Soc. 106, 1993. MR 94d:53114

[K]
D. Klain, Star valuations and dual mixed volumes, Adv. Math. 121 (1996), 80-101. MR 97i:52009

[L]
E. Lutwak, Intersection bodies and dual mixed volumes, Adv. Math. 71 (1988), 232-261. MR 90a:52023

[R1]
De-lin Ren, Topics in Integral Geometry, World Scientific Publishing, River Edge, NJ, 1994. MR 96h:53087

[R2]
-, Two topics in integral geometry, Proceedings of the 1981 symposium on differential geometry and differential equations (Shanghai-Hefei), Science Press, Beijing, 1984. MR 87d:53124

[San]
L.A. Santaló, Integral Geometry and Geometric Probability, Addison-Wesley, 1976. MR 55:6340

[Sch]
R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 1993. MR 94d:52007

[Z1]
G. Zhang, Centered bodies and dual mixed volumes, Trans. Amer. Soc. 345 (1994), 777-801. MR 95d:52008

[Z2]
-, Integral geometric inequalities, Acta Math. Sinica 34 (1991), 72-90. MR 92g:53066

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Additional Information:

Gaoyong Zhang
Affiliation: School of Mathematics, Institute for Advanced Study and Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
Address at time of publication: Department of Mathematics, Polytechnic University, 6 Metrotech Center, Brooklyn, New York 11201
Email: gzhang@math.poly.edu

DOI: 10.1090/S0002-9947-99-02053-X
PII: S 0002-9947(99)02053-X
Keywords: Kinematic formula, dual quermassintegral, chord power integral, dual mixed volume, star body, convex body, geometric probability
Received by editor(s): September 13, 1996
Received by editor(s) in revised form: December 2, 1996
Additional Notes: Research supported partially by NSF Grant DMS--9504913.
Dedicated: Dedicated to Professor De-lin Ren on the occasion of his 65th birthday
Copyright of article: Copyright 1999, American Mathematical Society

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