On the Minimization of Total $$(n-2)$$-th Mean Curvature in $$\mathbb {R}^n$$

Abstract This paper studies the Alexandrov–Fenchel inequality $$\\int _{\\Sigma } H_{n-2} d A \\ge \\omega _n^{\\frac{n-2}{n-1}} A(\\Sigma )^{\\frac{1}{n-1}}$$ ∫ Σ H n - 2 d A ≥ ω n n - 2 n - 1 A ( Σ ) 1 n - 1 for $$C^{1,1}$$ C 1 , 1 hypersurfaces $$\\Sigma \\subset \\mathbb {R}^n$$ Σ ⊂ R n , where $$\\omega _n$$ ω n is the surface area of $$\\mathbb {S}^{n-1}$$ S n - 1 . We prove this inequality for axiconvex hypersurfaces and extend it to axisymmetric cases. For any axisymmetric $$\\Sigma $$ Σ , we derive an asymptotic estimate for $$\\frac{1}{\\operatorname {diam}(\\Sigma )} \\int _{\\Sigma } H_{n-2} d A-\\frac{\\omega _{n-1}}{n-1}$$ 1 diam ( Σ ) ∫ Σ H n - 2 d A - ω n - 1 n - 1 . Moreover, we prove a stability estimate for the mean width-diameter inequality of convex bodies.