Weighted Alexandrov-Fenchel type inequalities for hypersurfaces in \(\mathbb{R}^n\) (Q6593638)

Let \(\Sigma\) be a closed hypersurface of class \(C^2\) in \({\mathbb R}^n\) (\(n\ge 3\)), and let \(H_j\) be the \(j\)th normalized elementary symmetric function of its principle curvatures (\(j=0,\dots,n-1\)). For convex \(\Sigma\), the inequalities between quermassintegrals (a special case of the Aleksandrov-Fenchel inequalities for mixed volumes) can be written as \N\[\N\left(\frac{1}{\omega_{n-1}}\int_\Sigma H_k\,d\mu\right)^{\frac{1}{n-1-k}} \ge \left(\frac{1}{\omega_{n-1}}\int_\Sigma H_l\,d\mu\right)^{\frac{1}{n-1-l}},\quad 0\le l<k\le n-1,\N\]\Nwhere \(\omega_{n-1}\) is the area of the unit sphere and \(\mu\) denotes the usual area measure on \(\Sigma\). The paper proves the following weighted generalization. Let \(\Sigma\) be star-shaped and \(k\)-convex (that is, \(H_j\ge 0\) for \(j=1,\dots,k\)), let \(0\le l<k\le n-1\) and \(\alpha\in[1,\infty)\). For \(x\in\Sigma\), let \(r\) be the distance of \(x\) from the origin. Then \N\[\N\left(\frac{1}{\omega_{n-1}}\int_\Sigma r^{2\alpha} H_k\,d\mu\right)^{\frac{1}{n-1-k+2\alpha}} \ge \left(\frac{1}{\omega_{n-1}}\int_\Sigma H_l\,d\mu\right)^{\frac{1}{n-1-l}},\quad 0\le l<k\le n-1,\N\]\Nwith equality if and only if \(\Sigma\) is a sphere centered at the origin. For \(\alpha=1\) this was proved by \textit{Y. Wei} and \textit{T. Zhou} [Bull. Lond. Math. Soc. 55, No. 1, 263--281 (2023; Zbl 1529.53086)]. The inequality is also known to be true for \(\alpha=0\) and \(\alpha=1/2\), and it is asked whether it holds for \(\alpha\in[0,1]\). The proof of the weighted inequality uses an inverse curvature flow argument. From the inequality it follows that the radius \(R_0\) of the smallest ball with center at the origin containing \(\Sigma\) (which is assumed to be \(k\)-convex) satisfies \N\[\NR_0\ge \left(\frac{\int_\Sigma H_l\,d\mu}{\omega_{n-1}}\right)^{\frac{1}{n-l-1}},\quad l=0,\dots,k-1.\N\]