The sufficient condition for a convex body to enclose another in $\textbf {R}^ 4$
HTML articles powered by AMS MathViewer
- by Jia Zu Zhou PDF
- Proc. Amer. Math. Soc. 121 (1994), 907-913 Request permission
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 ${\mathbb {R}^4}$.References
- Luis A. Santaló, Integral geometry and geometric probability, Encyclopedia of Mathematics and its Applications, Vol. 1, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. With a foreword by Mark Kac. MR 0433364
- Shiing-shen Chern, On the kinematic formula in the Euclidean space of $n$ dimensions, Amer. J. Math. 74 (1952), 227–236. MR 47353, DOI 10.2307/2372080 Delin Ren, Introduction to Integral Geometry, Shanghai Press of Sciences and Technology, 1987.
- H. Hadwiger, Überdeckung ebener Bereiche durch Kreise und Quadrate, Comment. Math. Helv. 13 (1941), 195–200 (German). MR 4995, DOI 10.1007/BF01378060
- H. Hadwiger, Gegenseitige Bedeckbarkeit zweier Eibereiche und Isoperimetrie, Vierteljschr. Naturforsch. Ges. Zürich 86 (1941), 152–156 (German). MR 7274
- Chang-shing Chen, On the kinematic formula of square of mean curvature vector, Indiana Univ. Math. J. 22 (1972/73), 1163–1169. MR 313977, DOI 10.1512/iumj.1973.22.22096
- Bang-yen Chen, Geometry of submanifolds, Pure and Applied Mathematics, No. 22, Marcel Dekker, Inc., New York, 1973. MR 0353212
- Shiing-shen Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math. (2) 45 (1944), 747–752. MR 11027, DOI 10.2307/1969302
- Gao Yong Zhang, A sufficient condition for one convex body containing another, Chinese Ann. Math. Ser. B 9 (1988), no. 4, 447–451. A Chinese summary appears in Chinese Ann. Math. Ser. A 9 (1988), no. 5, 635. MR 998651 Jiazu Zhou, Analogues of Hadwiger’s theorem in space ${\mathbb {R}^n}$ and sufficient conditions for a convex domain to enclose another, submitted. —, Generalizations of Hadwiger’s theorem and sufficient conditions for a convex domain to fit another in ${\mathbb {R}^3}$, submitted. —, A kinematic formula and analogues of Hadwiger’s theorem in space, Contemp. Math., vol. 140, Amer. Math. Soc., Providence, RI, 1992, pp. 159-167. —, When can one domain enclose another in space, J. Austral. Math. Soc. Ser. A (to appear). —, Kinematic formulas for the power of mean curvature and Hadwiger’s theorem in space, Trans. Amer. Math. Soc. (to appear).
- P. R. Goodey, Connectivity and freely rolling convex bodies, Mathematika 29 (1982), no. 2, 249–259 (1983). MR 696880, DOI 10.1112/S002557930001233X
- P. R. Goodey, Homothetic ellipsoids, Math. Proc. Cambridge Philos. Soc. 93 (1983), no. 1, 25–34. MR 684271, DOI 10.1017/S0305004100060291
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 121 (1994), 907-913
- MSC: Primary 52A22; Secondary 53C65, 60D05
- DOI: https://doi.org/10.1090/S0002-9939-1994-1184090-4
- MathSciNet review: 1184090