The Willmore functional and the containment problem in \(R^{4}\) (Q885566)

Let \(M\) be a closed surface in the Euclidean space \(\mathbb R^3\) with the mean curvature \(H\). Then the integral of the square mean curvature of \(M\) is called the Willmore functional. The known result of Willmore is \(\int_MH^2\,d\sigma\geqslant4\pi\), where \(d\sigma\) is the volume element of \(M\), with equality if and only if \(M\) is a standard sphere. Using the method of integral geometry, by applying the theory of kinematic formulas and Minkowski geometry, the author estimates the Willmore functional for a 3-dimensional convex hypersurface \(M\) in \(\mathbb R^4\). The integral \(\int_M H^2\,d\sigma\) has a sharp lower bound expressed by the surface area of \(M\), the volume of convex body and the Minkowski quermassintegrals of \(M\). The author considers also the containment problem that is closely related to the isoperimetric inequality. The containment problem asks: Given domains \(D_k\) \((k = i,j)\). When one domain can be ``moved'' inside another? Roughly speaking, when \(D_i\) can contain a congruent copy of \(D_j\). Generally we wish to have an answer that depends only upon the geometric invariants of the domains involved. In 1942 Hadwiger gave some sufficient conditions to insure that a planar domain \(D_i\) can contain a copy of \(D_j\). But the analogue of Hadwiger's containment problem for higher dimensions is still unknown for many cases though some results have been achieved. In this paper, the author obtains a new analogue of Hadwiger's condition for the convex bodies in \(\mathbb R^4\).