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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Kinematic formulas for mean curvature powers of hypersurfaces and Hadwiger's theorem in $ {\bf R}\sp {2n}$


Author: Jia Zu Zhou
Journal: Trans. Amer. Math. Soc. 345 (1994), 243-262
MSC: Primary 52A22; Secondary 51M16
DOI: https://doi.org/10.1090/S0002-9947-1994-1250829-7
MathSciNet review: 1250829
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Abstract: We first discuss the theory of hypersurfaces and submanifolds in the m-dimensional Euclidean space leading up to high dimensional analogues of the classical Euler's and Meusnier's theorems. Then we deduce the kinematic formulas for powers of mean curvature of the $ (m - 2)$-dimensional intersection submanifold $ {S_0} \cap g{S_1}$ of two $ {C^2}$-smooth hypersurfaces $ {S_0}$, $ {S_1}$, i.e., $ {\smallint _G}({\smallint _{{S_0} \cap g{S_1}}}{H^{2k}}d\sigma )dg$. Many well-known results, for example, the C-S. Chen kinematic formula and Crofton type formulas are easy consequences of our kinematic formulas. As direct applications of our formulas, we obtain analogues of Hadwiger's theorem in $ {\mathbb{R}^{2n}}$, i.e., sufficient conditions for one domain $ {K_\beta }$ to contain, or to be contained in, another domain $ {K_\alpha }$.


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DOI: https://doi.org/10.1090/S0002-9947-1994-1250829-7
Keywords: Mean curvature, normal curvature, principal curvature, kinematic density, kinematic formula, kinematic measure, domain, convex body
Article copyright: © Copyright 1994 American Mathematical Society