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The sufficient condition for a convex body to enclose another in $ {\bf R}\sp 4$

Author: Jia Zu Zhou
Journal: Proc. Amer. Math. Soc. 121 (1994), 907-913
MSC: Primary 52A22; Secondary 53C65, 60D05
MathSciNet review: 1184090
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Abstract: We follow Hadwiger and Ren's ideas to estimate the kinematic measure of a convex body $ {D_1}$ with $ {C^2}$-boundary $ \partial {D_1}$ moving inside another convex body $ {D_0}$ with the same kind of boundary $ \partial {D_0}$ under the isometry group G in $ {\mathbb{R}^4}$. By using Chern and Yen's kinematic fundamental formula, C-S. Chen's kinematic formula for the total square mean curvature $ {\smallint _{\partial {D_0} \cap g\partial {D_1}}}{H^2}dv$, and some well-known results about the curvatures of the 2-dimensional intersection submanifold $ \partial {D_0} \cap g\partial {D_1}$, we obtain a sufficient condition to guarantee that one convex body can enclose another in $ {\mathbb{R}^4}$.

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Keywords: Kinematic density, kinematic formula, kinematic measure, convex body, domain, mean curvature, scalar curvature
Article copyright: © Copyright 1994 American Mathematical Society