A note on new weighted geometric inequalities for hypersurfaces in \(\mathbb{R}^n\) (Q6583774)

Let \(\Omega\) be a smooth bounded domain in \(\mathbb{R}^n\) with boundary \(\partial \Omega=\Sigma\). The classical isoperimetric inequality is \N\[\frac{|\Sigma|}{\omega_{n-1}}\ge \left(\frac{\mathrm{Vol}(\Omega)}{\frac{\omega_{n-1}}{n}}\right)^{\frac{n-1}{n}},\] \Nwhere \(|\Sigma|\) denotes the area of \(\Sigma\) and \(\omega_{n-1}\) is the area of the unit sphere \(S^{n-1}\). Let \(\kappa = (\kappa_1, \kappa_2, \dots, \kappa_{n-1})\) be the principal curvatures of \(\Sigma\) and for any integer \(1 \le k \le n - 1\), define the \(k\)-th mean curvature \(\sigma_k\) and the normalized \(k\)-th mean curvature \(H_k\) of \(\Sigma\) by \[\sigma_k = \sum_{1\le i_1<\dots< i_k\le n-1} \kappa_{i_1}\dots \kappa_{i_k},\] \[H_k = {\frac{k!\, \sigma_k}{(n-1) (n-2)\dots(n-k)}},\] \(\sigma_0 =H_0=1\) and \(\sigma_k= H_k = 0\) for \(k\ge n\). Let \(r\) be the Euclidean distance to some given point \(O\in \mathbb{R}^n\). \(\Omega\) in \(\mathbb{R}^n\) is called star-shaped, if its support function \(u = \langle X, \nu\rangle\) is positive everywhere on \(\Sigma\), where \(X\) is the position function of \(\Sigma\) and \(\nu\) is the unit outer-normal of \(\Sigma\) at \(X\). A hypersurface is called \(k\)-convex if its principal curvatures satisfy \(H_i>0\) for all \(1 \le i \le k\). In particular, 1-convex is also called mean convex. \N\N\textit{K.-K. Kwong} and \textit{P. Miao} [Pac. J. Math. 267, No. 2, 417--422 (2014; Zbl 1295.53074)] first obtained a sharp inequality relating \(r^2\)-weighted mean curvature and the enclosed volume for a convex hypersurface and then extended it to a star-shaped, mean-convex hypersurface through the inverse mean curvature flow. For higher order mean curvature integrals, \textit{K.-K. Kwong} and \textit{P. Miao} [Commun. Contemp. Math. 17, No. 5, Article ID 1550014, 10 p. (2015; Zbl 1325.53086)] also established a family of weighted inequalities. \N\NThe main result of the paper is the following statement. \N\NTheorem. Let \(\Sigma\) be a smooth, closed, star-shaped and \(k\)-convex hypersurface in \(\mathbb{R}^n\) (\(n\ge 3\)). Then for each \(k = 1, \dots, n-1\), one has the inequality \N\[\N\int_{\Sigma} r^2 H_k d\mu + {\frac{2 (k-1)}{n-k+1}} \int_{\Sigma} H_{k-2} d\mu \ge {\frac{n+k-1}{n-k+1}} \omega_{n-1} \left({\frac{\int_{\Sigma}H_{k-1}\,d\mu}{\omega_{n-1}}}\right)^{{\frac{n-k+1}{n-k}}}.\N\]