The sufficient condition for a convex body to enclose another in 𝑅⁴

Abstract

We follow Hadwiger and Ren’s ideas to estimate the kinematic measure of a convex body D 1 {D_1} with C 2 {C^2} -boundary ∂ D 1 \partial {D_1} moving inside another convex body D 0 {D_0} with the same kind of boundary ∂ D 0 \partial {D_0} under the isometry group G in R 4 {\mathbb {R}^4} . By using Chern and Yen’s kinematic fundamental formula, C-S. Chen’s kinematic formula for the total square mean curvature ∫ ∂ D 0 ∩ g ∂ D 1 H 2 d v {\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 ∂ D 0 ∩ g ∂ D 1 \partial {D_0} \cap g\partial {D_1} , we obtain a sufficient condition to guarantee that one convex body can enclose another in R 4 {\mathbb {R}^4} .