Alexandrov--Fenchel inequalities for closed axisymmetric hypersurfaces

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 $${\left(\frac{{\int }_{\mathrm{\Sigma }}{H}_{\mathrm{\ell }}dA}{\mathrm{\mid }{\mathbb{S}}^n\mathrm{\mid }}\right)}^{1\mathrm{/}(n-\mathrm{\ell })}\le {\left(\frac{{\int }_{\mathrm{\Sigma }}{H}_{k}dA}{\mathrm{\mid }{\mathbb{S}}^n\mathrm{\mid }}\right)}^{1\mathrm{/}(n-k)},0\le \mathrm{\ell }<k\le n-1.$$ Without axiconvexity, the inequalities hold for sign-corrected integrals $${\int }_{\mathrm{\Sigma }}{\epsilon }_{\theta }^j{H}_{j}dA,$$ 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.