## 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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