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Hyperbolic Alexandrov-Fenchel quermassintegral inequalities. II. (English) Zbl 1301.53077

Summary: In this paper we first establish an optimal Sobolev-type inequality for hypersurfaces in \(\mathbb H^n\) (see Theorem 1.1). As an application we obtain hyperbolic Alexandrov-Fenchel inequalities for curvature integrals and quermassintegrals. More precisely, we prove a geometric inequality in the hyperbolic space \(\mathbb H^n\), which is a hyperbolic Alexandrov-Fenchel inequality,
\[ \int_\Sigma \sigma_{2k}\geq C^{2k}_{n-1}\omega_{n-1}\left\{\left(\frac{|\Sigma|}{\omega_{n-1}}\right)^{\frac{1}{k}}+\left(\frac{|\Sigma|}{\omega_{n-1}}\right)^{\frac{1}{k}\frac{n-1-2k}{n-1}}\right\}^k, \]
when \(\Sigma\) is a horospherical convex and \(2k\leq n-1\). Equality holds if and only if \(\Sigma\) is a geodesic sphere in \(\mathbb H^n\). Here \(\omega_j=\omega_j(\kappa)\) is the \(j\)th mean curvature and \(\kappa=(\kappa_1,\kappa_2,\cdots, \kappa_{n-1})\) is the set of the principal curvatures of \(\Sigma\). Also, an optimal inequality for quermassintegral in \(\mathbb H^n\) is \[ W_{2k+1}(\Omega)\geq\frac{\omega_{n-1}}{n}\sum\limits_{i=0}^k\frac{n-1-2k}{n-1-2k+2i}C_k^i\left(\frac{nW_1(\Omega)}{\omega_{n-1}}\right)^{\frac{n-1-2k+2i}{n-1}}, \] provided that \(\Omega\subset\mathbb H^n\) is a domain with \(\Sigma=\partial\Omega\) horospherical convex, where \(2k\leq n-1\). Equality holds if and only if \(\Sigma\) is a geodesic sphere in \(\mathbb H^n\). Here \(W_r(\Omega)\) is quermassintegrals in integral geometry.

MSC:

53C65 Integral geometry
52A40 Inequalities and extremum problems involving convexity in convex geometry
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