IJPAM: Volume 37, No. 4 (2007)

THE SUFFICIENT CONDITION FOR A CONVEX BODY
TO ENCLOSE ANOTHER IN A REAL SPACE FORM

Yoshio Matsuyama
Department of Mathematics
Chuo University
1-13-27 Kasuga, Bunkyo-ku, Tokyo, 112-8551, JAPAN
e-mail: matuyama@@math.chuo-u.ac.jp


Abstract.We follow Hadwiger and Ren's ideas (see [#!H!#], [#!H1!#], and [#!R!#]) 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 a real space form $\tilde{M}^{4}(c)$ of the constant curvature $c$. By using Chern and Yen's (see [#!C!#]) kinematic fundamental formula and C.-S. Chen's (see [#!C2!#]) kinematic formula for the total square mean curvature $\int_{\partial D_{0} \cup_{g} \partial D_{1}} H^{2} dv$ we obtain a sufficient condition to guarantee that one convex body can enclose another in $\tilde{M}^{4}(c)$.

Received: April 3, 2007

AMS Subject Classification: 52A22, 53C05, 51M16

Key Words and Phrases: kinematic formula, kinematic measure, convex body, domain, mean curvature

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 37
Issue: 4