Alexandrov--Fenchel inequalities for closed axisymmetric hypersurfaces

Abstract

We establish sharp Alexandrov--Fenchel inequalities for closed, embedded, axisymmetric hypersurfaces Sigma in n+1 dimensional Euclidean space R^{n+1}, the dimension n > or = 2, with admissible C^{1,1} profiles. Let H_j be the j-th normalized mean curvature. If Sigma is axiconvex, i.e., every hyperplane orthogonal to the symmetry axis meets the enclosed domain in an n-dimensional ball or empty, then \[\left(\frac{\int_ΣH_\ell\,dA}{|\mathbb S^n|}\right)^{1/(n-\ell)} \le \left(\frac{\int_ΣH_k\,dA}{|\mathbb S^n|}\right)^{1/(n-k)}, \qquad 0\le\ell<k\le n-1. \] Without axiconvexity, the inequalities hold for sign-corrected integrals \[\int_Σ\varepsilon_θ^{\,j}H_j\,dA,\] where varepsilon_{theta}= {sgn}(sin {theta}) denotes the sign of the azimuthal principal curvatures. As applications, the Alexandrov--Fenchel inequalities remain valid whenever both l and k are even. Equality in each of the above inequalities holds if and only if Sigma is a round sphere. The proof adapts the rearrangement and tangent angle folding approach of Dalphin-Henrot-Masnou-Takahashi from surfaces of revolution in R^3 to n-dimensional axisymmetric hypersurfaces in R^{n+1}. The higher-dimensional construction uses weighted monotone rearrangement to produce convex comparison bodies with C^{1,1} boundaries.